𝐙/𝐦-graded lie algebras and perverse sheaves,...

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REPRESENTATION THEORYAn Electronic Journal of the American Mathematical SocietyVolume 21, Pages 322–353 (September 15, 2017)http://dx.doi.org/10.1090/ert/501

Z/m-GRADED LIE ALGEBRAS AND PERVERSE SHEAVES, II

GEORGE LUSZTIG AND ZHIWEI YUN

Abstract. We consider a fixed block for the equivariant perverse sheaves withnilpotent support on the 1-graded component of a semisimple cyclically gradedLie algebra. We give a combinatorial parametrization of the simple objects inthat block.

Contents

Introduction 32210. The vector space V and the sesquilinear form (:) 32311. The A-lattice VA 33712. Purity properties 34213. An inner product 34314. Odd vanishing 348References 353

Introduction

As in [LY] we fix G, a semisimple simply connected algebraic group over k (analgebraically closed field of characteristic p ≥ 0) and a Z/m-grading g = ⊕i∈Z/mgi

for the Lie algebra g of G; here m is an integer > 0 and p is 0 or a large primenumber. We fix η ∈ Z − {0} and let δ be the image of η in Z/m. Let G0 bethe closed connected subgroup of G with Lie algebra g0; this group acts naturally

on gδ and on gnilδ = gδ ∩ gnil where gnil is the variety of nilpotent elements ing. Recall that I(gδ) denotes the set of pairs (O,L) where O is a G0-orbit on gnilδ

and L is an irreducible G0-equivariant local system on O up to isomorphism. LetB be the set of isomorphism classes of simple G0-equivariant perverse sheaves on

gnilδ . Then there is a natural bijection I(gδ) → B given by (O,L) �→ L�[dimO](for the notation L� see [LY, 0.11]). In [LY, Theorem 0.6] we have shown that B

can be naturally decomposed as a disjoint union of blocks ξB indexed by the G0-

conjugacy classes of admissible systems ξ = (M,M0,m,m∗, C) such that if B,B′

are in different blocks then the G0-equivariant Ext-groups of B with B′ are zero.This generalizes a result of [L4] in the Z-graded case; its analogue in the ungradedcase is the known partition of the set G-equivariant simple perverse sheaves on gnil

into blocks given by the generalized Springer correspondence.In this paper we give an (essentially) combinatorial way to parametrize the ob-

jects in a fixed block ξB of B. It is known that in the ungraded case, the objects in

Received by the editors October 12, 2016, and, in revised form, June 23, 2017.2010 Mathematics Subject Classification. Primary 20G99.The first author was supported by NSF grant DMS-1566618.The second author was supported by NSF grant DMS-1302071 and the Packard Foundation.

c©2017 American Mathematical Society

322

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Z/m-GRADED LIE ALGEBRAS AND PERVERSE SHEAVES, II 323

a fixed block are indexed by the irreducible representations of a certain (relative)Weyl group attached to the block. In the Z/m-graded case we will associate tothe block ξB a Q-vector space E with a certain finite collection of hyperplanes.The complement of the union of these hyperplanes is naturally a union of finitelymany (not necessarily conical) chambers which can be taken to form a basis of aQ(v)-vector space V′ (here v is an indeterminate). The chambers represent variousspiral induction associated to the block. The vector space V′ carries a natural,explicit, sesquilinear form (:) : V′ ×V′ → Q(v) defined in terms of dimensions ofExt-groups between the spiral inductions (see [LY, 6.4]) which correspond to thechambers. We show that the left radical of this form is the same as its right radical.Taking the quotient of V′ by the left or right radical we obtain a Q(v)-vector spaceV with an induced sesquilinear form (:). It turns out that V is naturally isomorphicto the Grothendieck group based on ξB (tensored with Q(v)) and, in particular,the number of elements in ξB is equal to dimQ(v) V. The vector space V has anatural bar involution¯ : V → V and a natural A-lattice VA in V, both definedcombinatorially. (Here A = Z[v, v−1].) We then define a subset B′ of V as the setof all b ∈ VA such that b = b and (b : b) ∈ 1+ vZ[[v]]. We show that B′ is a signedbasis of V. (Although the definition of B′ is combinatorial, the proof of the factthat it is a well-defined signed basis is based on geometry; it is not combinatorial.It would be desirable to find a proof without using geometry.) Let B′/± be theset of orbits of the Z/2-action b �→ −b on B′. We show that B′/± is in naturalbijection with the given block ξB. Thus B′/± could be regarded as a combinatorialindex set for ξB. (A similar result in the Z-graded case appears in [L4].)

We now discuss the contents of the various sections. In Section 10 we define theQ-vector vector E with its hyperplane arrangement associated to a block. We alsodefine the Q(v)-vector space V′ with its sesquilinear form (:), its quotient space Vand the bar operator. In Section 11 we define the A-lattice VA in V and the signedbasis B′. In Section 12 we prove some purity properties of the cohomology sheavesof the simple G0-equivariant perverse sheaves on gnilδ , which generalize those in theZ-graded case given in [L4]. In Section 13 we generalize an argument in [L4] toexpress the matrix whose entries are the values of the (:)-pairing at two elements ofB as a product of three matrices. This is used in Section 14 to prove the vanishingof the odd cohomology sheaves of the intersection cohomology of the closure of anyG0-orbit in gnilδ with coefficients in an irreducible G0-equivariant local system onthat orbit. This generalizes a result in [L4] in the Z-graded case whose proof wasquite different (it was based on geometric arguments which are not available in thepresent case).

We will adhere to the notation and assumptions of [LY].

10. The vector space V and the sesquilinear form (:)

In this section, we introduce a combinatorial way of calculating the number ofirreducible perverse sheaves in a block ξQ(gδ). This is achieved by introducing afinite-dimensional vector space V′ over Q(v) together with a sesquilinear form onit coming from the pairing between spiral inductions in the fixed block ξ. Then thenumber of irreducible perverse sheaves in ξQ(gδ) turns out to be the rank of thissesquilinear form on V′ (Proposition 10.19).


324 GEORGE LUSZTIG AND ZHIWEI YUN

10.1. We fix ξ ∈ Tη and a representative ξ = (M,M0,m,m∗, C) ∈ Tη for ξ. We also

fix φ = (e, h, f) ∈ JM such that e ∈ ◦mη, h ∈ m0, f ∈ m−η. We set ι = ιφ ∈ YM .

LetZ = Z0

M .

Since Z is a torus, YZ is naturally a free abelian group (with operation writtenas addition) and E := YZ,Q may be naturally identified with the Q-vector spaceQ⊗ YZ . Let XZ = Hom(YZ ,Z) and let 〈:〉 be the obvious perfect bilinear pairingYZ ×XZ → Z. This extends to a bilinear pairing E×(Q⊗XZ) → Q denoted againby 〈:〉. For any α ∈ XZ let

gα = {x ∈ g; Ad(z)x = α(z)x ∀z ∈ Z}.For any (α, i) ∈ XZ × Z/m let

gαi = g

α ∩ gi.

For any (α, n, i) ∈ XZ × Z× Z/m let

gα,ni = g

αi ∩ (ιngi).

For any i ∈ Z/m, let Ri = {(α, n) ∈ XZ ×Z; gα,ni �= 0}, R∗i = {(α, n) ∈ Ri;α �= 0}.

Note that(a) dim g

α,ni = dim g

−α,−n−i for any α, n, i; hence (α, n) �→ (−α,−n) is a bijection

Ri∼−→ R−i and a bijection R∗

i∼−→ R∗

−i.We have

g = ⊕(α,n,i)∈XZ×Z×Z/m(gα,ni ) = ⊕i∈Z/m,(α,n)∈Ri(gα,ni ).

For any N ∈ Z and any (α, n) ∈ R∗N we set

Hα,n,N = {� ∈ E; 〈� : α〉 = 2N/η − n}.This is an affine hyperplane in E. We set

E′ = E− ∪N∈Z,(α,n)∈R∗NHα,n,N .

10.2. Recall the notation YH,Q for a connected algebraic group H with Lie algebrah from [LY, 0.11]. For μ, μ′ ∈ YH,Q, we say μ commutes with μ′ if for some r, r′ inZ>0 such that rμ, r′μ′ ∈ YH , the images of the homomorphisms rμ, r′μ′ : k∗ → Hcommute with each other. This property is independent of the choice of r, r′. Ifμ commutes with μ′, and r, r′ ∈ Z>0 are as above, then letting λ = rμ, λ′ = r′μ′,we have the homomorphism ν : k∗ → H given by ν(t) = λ(tr

′)λ′(tr). We then

define μ + μ′ := ν/(rr′) ∈ YH,Q. Then μ + μ′ is independent of the choice of r, r′.Moreover, for any κ ∈ Q, we have

(a) μ+μ′

κ h = ⊕s,t∈Q;s+t=κ(μsh ∩

μ′

t h).

Let � ∈ E. Since � ∈ E = YZ,Q commutes with ι ∈ YM0, by the above discussion,

we may define

� =|η|2(� + ι) ∈ YM0,Q ⊂ YG0,Q.

From the definitions, � may be computed as follows. We choose f ∈ Z>0 such thatλ′ := f� ∈ YZ and f/|η| ∈ Z; we define λ ∈ YG0

by λ(t) = ι(tf )λ′(t) = λ′(t)ι(tf )

for all t ∈ k∗; we then have � = |η|2f λ ∈ YG0,Q.

We shall set ε = η ∈ {1,−1} (the sign of η, see 0.12).

Now εp�∗ , εl

�∗ are well-defined; see [LY, 2.5, 2.6].



We setεl� = ⊕N∈Z

εl�N .

We show:(b) We have εl

�∗ = m∗ if and only if � ∈ E′.

If x ∈ gα,ni and t ∈ k∗, then

Ad(λ(t))x = Ad(ι(tf )) Ad(λ′(t))x = tnf+〈λ′:α〉x

and Ad(ι(t))x = tnx. Thus, gα,ni ⊂ λnf+〈λ′:α〉gi and g

α,ni ⊂ ι

ngi. It follows that for

any k ∈ Z we have

(c) λkgi = ⊕(α,n)∈Ri;nf+〈λ′:α〉=k(g

α,ni ),

(d) ιkgi = ⊕(α,n)∈Ri;n=k(g

α,ni ).

For any N ∈ Z we have εl�N = λ

2fN/ηgN . We see that

(e) εl�N = ⊕(α,n)∈RN ;nf+〈λ′:α〉=2fN/η(g

α,nN ).

Recall from 1.2(e) that if N ∈ Z,mN �= 0, then N ∈ ηZ. Let N ∈ ηZ. Fromthe arguments in 3.3 (or by [LY, 10.3(a)]) we see that, for some �′ ∈ E, we haveεl�′

∗ = m∗. (Here �′ is attached to �′ in the same way that � was attached to �.)Hence, using (e) with � replaced by �′, we see that there exists a subset RN of RN

such that mN = ⊕(α,n)∈RN(gα,nN ). If (α, n) ∈ RN , then 0 �= g

α,nN ⊂ mN . Hence for

x ∈ gα,nN − {0} and z ∈ Z, t ∈ k∗, we have Ad(z)x = α(z)x, Ad(ι(t))x = tnx; but

Ad(z)x = x since Z is in the center of M and Ad(ι(t))x = t2N/ηx since x ∈ mN .Thus α(z) = 1 (so that α = 0) and n = 2N/η. We see that RN ⊂ {(0, 2N/η)}.Conversely, we show that g

0,2N/ηN ⊂ mN . If x ∈ g

0,2N/ηN , then Ad(z)x = x for all

z ∈ Z and Ad(ι(t))x = t2N/ηx for all t ∈ k∗. Since ξ in 9.1 is admissible, thereexists t0 ∈ k∗, z ∈ Z, such that m is the fixed point set of Ad(ι(t0)z)θ : g → g.Since mη is contained in this fixed point set and Ad(z) acts trivially on it, we seethat t20ζ

ηy = y for all y ∈ mη.

If mη �= 0, it follows that t20ζη = 1. Thus we have Ad(ι(t)z)θ(x) = t

2N/η0 ζNx =

(t20ζη)N/ηx = x. (Here we use that N ∈ ηZ.) Thus x is contained in the fixed point

set of Ad(ι(t0)z)θ : g → g, so that x ∈ mN . We see that for any N ∈ ηZ we have

(f) mN = g0,2N/ηN .

Now (f) also holds when mη = 0. (In that case we must have ι = 1 hence m = m0,so that mN = 0 for N �= 0. Moreover, by [LY, 3.6(d)], m is a Cartan subalgebra of

g0 and Z = M . We also have g0,2N/ηN = 0 for N �= 0. Thus, for N �= 0, (f) states

that 0 = 0. If N = 0, (f) states that m is its own centralizer in g0, which is clear.)

From (e),(f) we see that mN ⊂ εl�N for all N ∈ ηZ and that

(g) for N ∈ ηZ we have mN = εl�N if and only if the following holds: {(α, n) ∈

RN ;n + 〈� : α〉 = 2N/η} is equal to {(0, 2N/η)} if (0, 2N/η) ∈ RN and is emptyif (0, 2N/η) /∈ RN ;

(g′) for N ∈ Z − ηZ we have mN = εl�N (or equivalently εl

�N = 0) if and only if

{(α, n) ∈ RN ;n+ 〈� : α〉 = 2N/η} = ∅. The condition in (g) can be also expressed



as follows:{(α, n) ∈ R∗

N ;n+ 〈� : α〉 = 2N/η} = ∅ for any N ∈ ηZ.

The condition in (g′) can be also expressed as follows:{(α, n) ∈ RN ;n+ 〈� : α〉 = 2N/η} = ∅ for any N ∈ Z− ηZ.Indeed, it is enough to show that if N ∈ Z−ηZ and {(α, n) ∈ RN ;n+ 〈� : α〉 =

2N/η}, then we have automatically α �= 0. (Assume that α = 0. Then n = 2N/η

so that n is an odd integer. Since g0 = m and g0,nN �= 0 we see that ι

nm �= 0. Using

1.2(d) we deduce that n is even, a contradiction.)We see that (b) holds.From (c) we deduce that

εp�N = ⊕k∈Z;k≥2fN/η(

λkgN ) = ⊕k∈Z,(α,n)∈RN ;k≥2fN/η,nf+〈λ′:α〉=k(g

α,nN ),

hence

(h) εp�N = ⊕(α,n)∈RN ;〈�:α〉≥2N/η−n(g

α,nN ).

The nilradical εu�∗ of εp

�∗ is given by

(i) εu�N = ⊕(α,n)∈R∗

N ;〈�:α〉>2N/η−n(gα,nN ).

We see that for �,�′ in E and N ∈ Z, the following two conditions are equivalent:

(I) εp�N = εp

�′

N .(II) For any (α, n) ∈ R∗

N we have 〈� : α〉+ n ≥ 2N/η ⇔ 〈�′ : α〉+ n ≥ 2N/η.

For �,�′ in E′ we say that � ≡ �′ if for any N ∈ Z and any (α, n) ∈ R∗N we

have

(〈� : α〉+ n− 2N/η)(〈�′ : α〉+ n− 2N/η) > 0.

This is clearly an equivalence relation on E′. From the equivalence of (I),(II) above,we see that

(j) � ≡ �′ ⇔ εp�N = εp

�′

N ∀N ∈ Z.

For any � ∈ E′ we set

(k)I� = ε Indgδ

εp�η(C[− dimmη]) ∈ Q(gδ),

I� = εIndgδ

εp�η(C) ∈ Q(gδ).

Here we regard C as an object of Q(l�η ) = Q(mη), see (b). Note that in K(gδ) we

have

I� = vh(�)I�,

where

h(�) = dim εu�0 + dim εu�η + dimmη = dim εu

�0 + dim εp�η .

We show:(l) If �,�′ ∈ E′, � ≡ �′, then I� = I�′ , h(�) = h(�′).

Indeed, in this case we have εp�N = εp

�′

N for all N ∈ Z (see (j)) and the resultfollows from the definitions.



10.3. We keep the setup of 10.1, 10.2. As in [LY, 2.9], for any N ∈ Z we set

lφN = ι

2N/ηgN if 2N/η ∈ Z, lφN = 0 if 2N/η /∈ Z.

Hence

(a) lφN = ⊕(α,n)∈RN ;n=2N/ηg

α,nN .

We set lφ = ⊕N∈Z lφN .

Let

E′′ = E− ∪N∈Z,(α,n)∈R∗N ;n=2N/ηHα,n,N .

For � ∈ E we show:(b) We have εl� ⊂ lφ if and only if � ∈ E′′.

Using (a) and 10.2(e) we see that we have ⊕Nεl�N ⊂ lφ if and only if for any

N ∈ Z we have

{(α, n) ∈ RN ;n+ 〈� : α〉 = 2N/η} ⊂ {(α, n) ∈ RN ;n = 2N/η},

or equivalently,

{(α, n) ∈ RN ;n+ 〈� : α〉 = 2N/η;n �= 2N/η} = ∅,

or equivalently,

{(α, n) ∈ R∗N ; 〈� : α〉 = 2N/η − n �= 0} = ∅.

This is the same as the condition that � ∈ E′′. This proves (b).

10.4. We show:(a) Let p∗ be an ε-spiral with a splitting m′

∗ such that m is a Levi subalgebra of aparabolic subalgebra of m′ = ⊕Nm′

N compatible with the Z-gradings. Then for some

� ∈ E we have p∗ = εp�∗ , m′

∗ = εl�∗ .

We can find μ ∈ YG0,Q such that p∗ = εpμ∗ and m′

∗ = εlμ∗ . Let M ′

0 = em′0 .

We choose f ∈ Z>0 such that λ1 := fμ ∈ YG0. We have m′

N = λ1

fNεgN for all

N ∈ Z. Let N ∈ ηZ. Since mN ⊂ m′N , for any x ∈ mN and any t ∈ k∗ we have

Ad(λ1(t))x = tfNεx; we have also Ad(ι(t))x = t2N/ηx.Hence Ad(λ1(t

2)ι(t−f |η|))x = x for any x ∈ mN . Since this holds for anyN ∈ ηZ,it follows that the image of λ2

1ι−f |η| : k∗ → M ′

0 commutes with M . Since M is aLevi subgroup of a parabolic subgroup of M ′, the image of λ2

1ι−f |η| is contained in

ZM , hence in Z0M = Z. In particular, the images of ι and λ2

1ι−f |η| commute with

each other, hence the images of λ1 and ι commute with each other. It thereforemakes sense to write λ′ := 2λ1 − f |η|ι ∈ YM and we actually have λ′ ∈ YZ . Let� = |η|−1f−1λ′ ∈ YZ,Q = E. We have � = |η|−1(2μ− |η|ι) hence �+ ι = 2|η|−1μ,that is, μ = � and (a) is proved.

Let C′ be the collection of ε-spirals p∗ such that m∗ is a splitting of p∗. We show:(b) C′ coincides with the collection of ε-spirals of the form εp

�∗ with � ∈ E′.

Assume first that p∗ ∈ C′. Using (a) with m′∗ = m∗ we see that for some � ∈ E

we have p∗ = εp�∗ , m∗ = εl

�∗ . Using 10.2(b), we see that � ∈ E′.

Conversely, assume that p∗ = εp�∗ for some � ∈ E′. From 10.2(b) we have

m∗ = εl�∗ . Thus p∗ ∈ C′. This proves (b).



Let C′′ be the collection of ε-spirals p∗ with the following property: there exists

a splitting m′∗ of p∗ such that mN ⊂ m′

N ⊂ lφN for all N . We show:

(c) C′′ coincides with the collection of ε-spirals of the form εp�∗ with � ∈ E′′.

Assume first that p∗ ∈ C′′ and let m′∗ be a splitting of p∗ as in the definition of C′′.

Now m is a Levi subalgebra of a parabolic subalgebra of m′ = ⊕Nm′N compatible

with the Z-gradings of m and m′. (Indeed, from the proof of 3.7(c) we see that

there exists λ ∈ YZ such that m = {y ∈ lφ; Ad(λ(t))y = y ∀t ∈ k∗}. Since m′ ⊂ lφ

we see that m = {y ∈ m′; Ad(λ(t))y = y ∀t ∈ k∗}, as required.) Using (a), we see

that for some � ∈ E we have p∗ = εp�∗ , m′

∗ = εl�∗ . Since εl

�N ⊂ l

φN for all N , we see

from 10.3(b) that � ∈ E′′.

Conversely, assume that p∗ = εp�∗ for some � ∈ E′′. Let m′

∗ = εl�∗ . From

10.3(b) we see that εl�N ⊂ l

φN for all N . We also have mN ⊂ εl

�N for all N . Thus

p∗ ∈ C′′. This proves (c).

Note that:(d) If p∗ ∈ C′′, then the splitting m′

∗ in the definition of C′′ is in fact unique.Indeed, assume that m′

∗, m′∗ are splittings of p∗ such that mN ⊂ m′

N , mN ⊂ m′N

for all N . By [LY, 2.7] we can find u ∈ U0 (U0 as in [LY, 2.5]) such that Ad(u)m′∗ =

m′∗. In particular, we have Ad(u)m′

0 = m′0. This implies u = 1 since m′

0, m′0 are

both Levi subalgebras of p0 containing m0. Hence m′∗ = m′

∗.

10.5. Let p∗, p′∗ be two ε-spirals such that pN ⊂ p′N for all N . Let u∗, u

′∗ be their

nilradicals. Then we have u′N ⊂ uN for all N . In particular, u′N ⊂ pN for all N .We show:

(a) ⊕NpN/u′N is a parabolic subalgebra of l′ = ⊕Np′N/u′N .

Let P0 = ep0 , P ′0 = ep

′0 and let U0 = UP0

, U ′0 = UP ′

0. We have P0 ⊂ P ′

0, U′0 ⊂ U0.

Now p∗ = εpμ∗ and p′∗ = εp

μ′

∗ for some μ = λ/r, μ′ = λ′/r′, with λ, λ′ ∈ YG0and

r, r′ ∈ Z>0. We have λ(k∗) ⊂ P0, λ′(k∗) ⊂ P ′

0. We can find Levi subgroups L0,

L′0 of P0, P

′0 such that L0 ⊂ L′

0. By conjugating λ (resp. λ′) by an element of U0

(resp. U ′0) we can assume that λ ∈ ZL0

, λ′ ∈ ZL′0. Since ZL′

0⊂ ZL0

, we have

λ(t)λ′(t′) = λ′(t′)λ(t) for any t, t′ in k∗. Hence we have g = ⊕k,k′∈Q,i∈Z/m(μ,μ′

k,k′ gi),

where μ,μ′

k,k′ gi =μkgi ∩

μ′

k′gi. We have

pN = ⊕k,k′∈Q;k≥Nε(μ,μ′

k,k′ gN ),

p′N = ⊕k,k′∈Q;k′≥Nε(μ,μ′

k,k′ gN ),

u′N = ⊕k,k′∈Q;k′>Nε(μ,μ′

k,k′ gN ).

Since u′N ⊂ pN ⊂ p′N we see that

pN = ⊕k,k′∈Q;k≥Nε,k′≥Nε(μ,μ′

k,k′ gN ),

u′N = ⊕k,k′∈Q;k≥Nε,k′>Nε(μ,μ′

k,k′ gN ),

hence

pN/u′N∼= ⊕k,k′∈Q;k≥Nε,k′=Nε(

μ,μ′

k,k′ gN ).

This is a subspace of

p′N/u′N

∼= l′N := ⊕k,k′∈Q;k′=Nε(

μ,μ′

k,k′ gN ).



Since μ and μ′ commute with each other, it makes sense to define ν = μ −μ′ ∈ YL′

0,Q. Let l′ = ⊕N l′N ⊂ g. Then L′

0 acts on l′ by the Ad-action and ν

induces a Q-grading l′ = ⊕k1∈Qνk1l′. From the definitions we see that ⊕Np′N/uN =

⊕k1∈Q;k1≥0(νk1l′). Thus (a) holds.

A similar argument shows:(b) If m∗ is a splitting of p∗, the obvious map m = ⊕NmN → ⊕NpN/u′N defines

an isomorphism of m onto a Levi subalgebra of ⊕NpN/u′N .

10.6. Let �,�′ in E′. For any t ∈ Q such that 0 ≤ t ≤ 1 we set

�t := t� + (1− t)�′.

We assume that there is a unique hyperplane H of the form H = Hα0,n0,N0for some

(α0, n0, N0) with N0 ∈ Z, (α0, n0) ∈ R∗N0

such that �t ∈ H for some t = s; this s

is necessarily unique since �0 /∈ H. Note, however, that the triple (α0, n0, N0) isnot uniquely determined by H. We set �′′ = �s,

p∗ = εp�∗ , p′∗ = ε

p�′

∗ , u∗ = εu�∗ , u′∗ = ε

u�′

∗ ,

p′′∗ = ε

p�′′

∗ , l′′∗ = εl�′′

∗ , l′′ = ⊕N∈Z l′′N ,

P0 = ep0 , P ′0 = ep

′0 , P ′′

0 = ep′′0 .

We show:(a) For any N ∈ Z we have pN ⊂ p′′N ; hence P0 ⊂ P ′′

0 . For any N ∈ Z we havep′N ⊂ p′′N ; hence P ′

0 ⊂ P ′′0 .

Using 10.2(h), we see that to prove the first sentence in (a) it is enough toshow that for any (α, n) ∈ R∗

N such that 〈� : α〉 ≥ 2N/η − n we have also

〈�′′ : α〉 ≥ 2N/η − n. (Since � ∈ E′, we must have 〈� : α〉 > 2N/η − n. Sincefor t ∈ Q, s < t ≤ 1 we have �t ∈ E′, it follows that for all such t we have〈�t : α〉 > 2N/η− n. Taking the limit as t �→ s we obtain 〈�′′ : α〉 ≥ 2N/η− n, asrequired). The second sentence in (a) is proved in a similar way.

We show:(b) Fix N ∈ Z. If H �= Hα,n,N for any (α, n) ∈ R∗

N , then we have pN = p′′N = p′N .

We first show that pN = p′′N . By the equivalence of (I),(II) in 10.2, it is enoughto show that for any (α, n) ∈ R∗

N we have

〈� : α〉+ n− 2N/η > 0 ⇔ 〈�′′ : α〉+ n− 2N/η > 0,

or equivalently, that c1cs > 0, where ct = 〈�t : α〉+ n− 2N/η for t ∈ Q such that0 ≤ t ≤ 1. From our assumptions we see that ct �= 0 for all t. It follows that eitherct > 0 for all t or ct < 0 for all t. In particular, c1cs > 0, as required. The proof ofthe equality p′N = p′′N is entirely similar.

We show:(c) If H �= Hα,n,0 for any (α, n) ∈ R∗

0 and H �= Hα,n,η for any (α, n) ∈ R∗δ , then

I� = I�′ , h(�) = h(�′).Indeed, in this case, by (b) we have pN = p′N for N ∈ {0, η} and the equality

I� = I�′ follows from the definitions. From p0 = p′0 we deduce that u0 = u′0. Usingthis and pη = p′η we deduce that h(�) = h(�′). This proves (c).

We show:(d) If H = Hα0,n0,0 for some (α0, n0) ∈ R∗

0 but H �= Hα,n,η for any (α, n) ∈ R∗δ ,

then I� ∼= I�′ , h(�) = h(�′).



By (a) we have pN ⊂ p′′N , p′N ⊂ p′′N for allN ∈ Z and by (b) we have pη = p′′η = p′η.We can now apply [LY, 4.5(b)] twice to conclude that

I� ∼= ⊕j∈JI�′′ [−2aj ], I�′ ∼= ⊕j′∈J′I�′′ [−2a′j′ ],

where aj (resp. a′j′) are integers such that

ρP ′′0 /P0!Ql = ⊕j∈JQl[−2aj ], ρP ′′

0 /P ′0!Ql = ⊕j′∈J′Ql[−2a′j′ ].

To show that I� ∼= I�′ it is then enough to show that

⊕j∈JQl[−2aj ] ∼= ⊕j′∈J′Ql[−2a′j′ ]

or that ρP ′′0 /P0!Ql

∼= ρP ′′0 /P ′

0!Ql. This is clear since P ′′

0 /P0, P′′0 /P

′0 are partial flag

manifolds of the reductive quotient L′′0 of P ′′

0 with respect to two associate parabolicsubgroups. The above argument shows also that dimUP0

= dimUP ′0hence dim u0 =

dim u′0. Moreover, from (b) we have pη = p′η; we see that h(�) = h(�′). This proves(d).

For N ∈ Z let qN (resp. q′N ) be the image of pN (resp. p′N ) under the obvious

projection p′′N → l′′N . From 10.5(a),(b), we see that q = ⊕NqN , q′ = ⊕Nq′N are

parabolic subalgebras of l′′ and m is a Levi subalgebra of both q and q′. We show:(e) If H = Hα0,2,η for some (α0, 2) ∈ R∗

δ , then I� ∼= I�′ and h(�) = h(�′).By (a) we have pN ⊂ p′′N , p′N ⊂ p′′N for all N ∈ Z. Since Hα0,2,η is the unique

hyperplane (as in 10.1) on which �′′ lies, we have �′′ ∈ E′′. Hence by 10.3(b)

we have l′′N ⊂ lφN for all N ∈ Z. In particular, the Z-grading of l′′ is η-rigid and

e ∈◦l′′η , so that

◦mη ⊂

◦l′′η . Let A ∈ Q(l′′η) be the simple perverse sheaf on l′′η such

that the support of A is l′′η and A| ◦mη

is equal up to shift to C| ◦mη

. Applying (twice)

[LY, 1.8(b)] and the transitivity formula 4.2(a) we deduce

ε Indgδpη(C) = ε Indgδ

p′′η(ind

l′′ηqη(C)) ∼= ⊕j

ε Indgδ

p′′η(A)[−2sj][dimmη − dim l

′′η ],

ε Indgδ

p′η(C) = ε Indgδ

p′′η(ind

l′′η

q′η(C)) ∼= ⊕j

ε Indgδ

p′′η(A)[−2sj][dimmη − dim l

′′η ],

where (sj) is a certain finite collection of integers. It follows that

ε Indgδpη(C) ∼= ε Indgδ

p′η(C).

This shows that I�, I�′ are isomorphic. It remains to show that h(�) = h(�′), forwhich it suffices to show the following two equalities:

(f) dim u0 = dim u′0,

(g) dim uη = dim u′η.

Since p0 and p′0 are associate parabolic subalgebras of p′′0 sharing the same Levisubalgebra m0, their unipotent radicals u0 and u′0 have the same dimension. Thisproves (f).

Now we show (g). By 10.2(i), we have

dim uη − dim u′η =

∑α∈XZ ;〈�:α〉>0,〈�′:α〉<0

dim gα,2δ −

∑α∈XZ ;〈�:α〉<0,〈�′:α〉>0

dim gα,2δ .

We want to show the above difference is zero. For this, it suffices to show thatdim g

α,2δ = dim g

−α,2δ for any α ∈ XZ . Note that g

α,2δ = lφη ∩ gα, which is the

α-weight space of Z on lφη . By the property of the sl2-action on lφ given by the



triple φ = (e, h, f), we have that ad(e) induces an isomorphism lφ−η

∼= lφη . Since Zcommutes with e, this isomorphism restricts to an isomorphism of α-weight spaces

lφ−η ∩ gα ∼= lφη ∩ gα, that is, g

α,−2−δ

∼= gα,2δ . Therefore dim g

α,2δ = dim g

α,−2−δ =

dim g−α,2δ , as desired. This proves (g) and finishes the proof of (e).

10.7. Let◦E = E− ∪(α,n)∈R∗

δ ;n=2Hα,n,η

= {� ∈ E; 〈� : α〉+ n− 2 �= 0 ∀(α, n) ∈ R∗δ such that n �= 2}.

Note that E′ ⊂ E′′ ⊂◦E. For ′�, ′′� in

◦E we say that ′� ∼ ′′� if for any (α, n) ∈ R∗

δ

such that n �= 2 we have

(〈′� : α〉+ n− 2)(〈′′� : α〉+ n− 2) > 0.

This is an equivalence relation on◦E. We show:

(a) Assume that ′�, ′′� in E′ satisfy ′� ∼ ′′�. Then I′�∼= I′′� and h(′�) =

h(′′�); hence I′�∼= I′′�.

By a known property of hyperplane arrangements, we can find a sequence�0, �1,. . . , �k in E′ such that�0 = ′�,�k = ′′� and such that for any j ∈ {0, 1, . . . , k−1}one of the following holds:

�j ≡ �j+1;�j = �,�j+1 = �′ are as in 10.6(c);�j = �,�j+1 = �′ are as in 10.6(d);�j = �,�j+1 = �′ are as in 10.6(e).Using 10.2(l) or 10.6(c) or 10.6(d) or 10.6(e) we see that for j ∈ {0, 1, . . . , k− 1}

we have I�j∼= I�j+1

and h(�j) = h(�j+1). This proves (a).

The set of equivalence classes on◦E for ∼ is denoted by

◦E; it is a finite set.

10.8. For �,�′ in E we set

(a)

τ (�,�′) =∑

(α,n)∈R∗δ ;(〈�:α〉+n−2)(〈�′:α〉+n−2)<0

dim gα,nδ ,

−∑

(α,n)∈R∗0 ;(〈�:α〉+n)(〈�′:α〉+n)<0

dim gα,n0 .

Using 10.2(i) we see that when �,�′ are in E′, then

(b) τ (�,�′) = − dimεu

�0 + εu

�′

0

εu�0 ∩ εu

�′

0

+ dimεu

�η + εu

�′

η

εu�η ∩ εu

�′η

.

10.9. We define Gφ,Mφ as in [LY, 3.6]. We show that:(a) The obvious map Mφ/M

0φ → (G0 ∩Gφ)/(G0 ∩Gφ)

0 is an isomorphism.

Recall that φ = (e, h, f) with e ∈ ◦mη. Let U (resp. U ′) be the unipotent radical

of G(e)0 (resp. (M ∩G(e))0). We have G(e) = GφU (semidirect product). Takingfixed point sets of ϑ we obtain G0∩G(e) = (G0∩Gφ)(G0∩U) (semidirect product).We have M ∩G(e) = MφU

′ (semidirect product). Taking fixed point set of ι(t) forall t ∈ k∗ we obtain M0 ∩G(e) = (M0 ∩Mφ)(M0 ∩U ′) = Mφ(M0 ∩U ′) (semidirectproduct). (We have used that Mφ ⊂ M0.) It follows that we have canonically

(M0 ∩G(e))/(M0 ∩G(e))0 = Mφ/M0φ,



(G0 ∩G(e))/(G0 ∩G(e))0 = (G0 ∩Gφ)/(G0 ∩Gφ)0.

It remains to use that

(M0 ∩G(e))/(M0 ∩G(e))0 = (G0 ∩G(e))/(G0 ∩G(e))0;

see [LY, 3.8(a)].

Recall from [LY, 3.6] that Z is a maximal torus of (Gφ ∩ G0)0. Let H be the

normalizer of Z in (Gφ ∩ G0)0. Let H ′ = {g ∈ G0; Ad(g)M = M,Ad(g)mk =

mk ∀k ∈ Z}. Note that M0 is a normal subgroup of H ′. Hence the groups H/Z,H ′/M0 are defined. We show:

(b) H ⊂ H ′.Let g ∈ (Gφ ∩G0)

0 be such that gZg−1 = Z. Let

(M ′,M ′0,m

′,m′∗) = (gMg−1, gM0g

−1,Ad(g)m,Ad(g)m∗).

Note that Z0M ′ = Z0

M = Z. Repeating the argument in the last paragraph in theproof of [LY, 3.6(a)], we see that

(M ′,M ′0,m

′,m′∗) = (M,M0,m,m∗).

(The argument is applicable since g ∈ Gφ ∩ G0.) We see that g ∈ H ′; this proves(b).

We show:(c) H ∩M0 = Z.From the injectivity of the map in (a) we see that Mφ∩(G0∩Gφ)

0 = M0φ. Hence

we have

M0 ∩ (G0 ∩Gφ)0 ⊂ (M0 ∩Gφ) ∩ (G0 ∩Gφ)

0 ⊂ Mφ ∩ (G0 ∩Gφ)0 = M0

φ,

so that

M0 ∩ (G0 ∩Gφ)0 ⊂ M0

φ.

The opposite inclusion is also true since Mφ ⊂ M0 and M0φ ⊂ G0 ∩ Gφ. It follows

that

M0 ∩ (G0 ∩Gφ)0 = M0

φ = Z.

The last equality is because e is distinguished in m. Now H ∩M0 is the normalizerof Z in M0∩(Gφ∩G0)

0, that is, the normalizer of Z in Z. We see that H∩M0 = Z.This proves (c).

We show:(d) H ′ = M0H.Since H ⊂ H ′, M0 ⊂ H ′, we have M0H ⊂ H ′. Now let g ∈ H ′. We show that

g ∈ M0H. Let φ′ = (Ad(g)e,Ad(g)h,Ad(g)f). We have

Ad(g)e ∈ ◦mη,Ad(g)h ∈ m0,Ad(g)f ∈ m−η.

Since both Ad(g)e, e are in◦mη, we can find g1 ∈ M0 such that Ad(g1) Ad(g)e = e.

Replacing g by g1g we can assume that we have Ad(g)e = e. Using [L4, 3.3] forJM , we can find g2 ∈ M0 such that

(Ad(g2) Ad(g)e,Ad(g2) Ad(g)h,Ad(g2) Ad(g)f) = (e, h, f).

We have g2g ∈ Gφ. Replacing g by g2g we can assume that g ∈ G0 ∩ Gφ. Usingthe surjectivity of the map in (a) we see that:

G0 ∩Gφ ⊂ Mφ(G0 ∩Gφ)0.



Thus we can write g in the form g3g′ with g3 ∈ Mφ, g

′ ∈ (G0 ∩Gφ)0. Replacing g

by g−13 g we see that we can assume that g ∈ (G0 ∩Gφ)

0. Since Ad(g)M = M , wesee that Ad(g)Z = Z. Thus g ∈ H. This proves (d).

From (b),(c),(d) we see that:

(e) The inclusion H ⊂ H ′ induces an isomorphism H/Z∼−→ H ′/M0. In particu-

lar, M0 is the identity component of H ′.

10.10. Let g ∈ H ′ (notation of 10.9). Then Ad(g) restricts to an isomorphism

mη∼−→ mη. Let C

′ = Ad(g)∗C, a simple perverse sheaf in Q(mη). We show:

(a) C ′ ∼= C.

Using 10.9(e) we can assume that g ∈ H (notation of 10.9). Since C, C ′ areintersection cohomology complexes attached to M0-equivariant irreducible localsystems on mη, they correspond to irreducible representations of

(M0 ∩G(e))/(M0 ∩G(e))0 = Mφ/M0φ.

Hence it is enough to show that Ad(g) induces the identity automorphism ofMφ/M

0φ. Using 10.9(a), we see that it is enough to show that Ad(g) induces

the identity automorphism of (G0 ∩ Gφ)/(G0 ∩ Gφ)0. This is obvious since g ∈

(G0 ∩Gφ)0. This proves (a).

10.11. Let �,�′ ∈ E′. Recall that P0 = eεp�0 ⊂ G0, P

′0 = e

εp�′0 ⊂ G0 are parabolic

subgroups of G0 with a common Levi subgroup M0. Let U0 = UP0, U ′

0 = UP ′0. Let

X be the set of all g ∈ G0 such that Ad(g)p�∗ and p

�′

∗ have a common splitting (asin 6.3). Note that X is a union of (P ′

0, P0)-double cosets in G0. We show:(a) We have H ′ ⊂ X (notation of 10.9). Let j : H ′/M0 → P ′

0\X/P0 be the mapinduced by the inclusion H ′ → X. Then j is a bijection.

If g ∈ H ′, then g ∈ G0 and Ad(g)m∗ = m∗ hence m∗ is a common splitting of

Ad(g)p�∗ and p

�′

∗ . Thus we have g ∈ X, so that the inclusionH ′ ⊂ X holds. Let g ∈X. Then g ∈ G0 and Ad(g)p

�∗ , p

�′

∗ have a common splitting m′∗. Then Ad(g)m∗,m

′∗

are splittings of Ad(g)p�∗ hence, by [LY, 2.7(a)], we have Ad(gug−1) Ad(g)m∗ = m′∗

for some u ∈ U0. Moreover, m∗,m′∗ are splittings of p

�′

∗ hence, by [LY, 2.7(a)], wehave Ad(u′)m∗ = m′

∗ for some u′ ∈ U ′0. It follows that Ad(gu)m∗ = Ad(u′)m∗ hence

u′−1gu ∈ H ′. Since U0 ⊂ P0, U′0 ⊂ P ′

0, we see that j is surjective. It remains toshow that j is injective. Let g, g′ be elements of H ′ such that g′ = p′0gp0 for somep0 ∈ P0, p

′0 ∈ P ′

0. We must only show that g′ ∈ gM0. Let NM0 be the normalizerof M0 in G0. It is enough to show that the obvious map NM0/M0 → P ′

0\G0/P0

is injective. This is a well-known property of parabolic subgroups and their Levisubgroups in a connected reductive group. This completes the proof of (a).

Let W = H/Z (notation of 10.9). Let w ∈ W and let g ∈ H be a representativeof w. Now Ad(g) restricts to an automorphism of Z which depends only on w;

this induces an isomorphism YZ∼−→ YZ and, by extension of scalars, a vector space

isomorphism E∼−→ E denoted by �1 �→ w�1. For any (α, n, i) ∈ XZ × Z × Z/m,

Ad(g) defines an isomorphism gα,ni

∼−→ gwα,ni where wα ∈ XZ is given by wα(z) =

α(w−1(z)); hence for any i, (α, n) �→ (wα, n) is a bijection Ri∼−→ Ri. Moreover, for

any N ∈ Z and any (α, n) ∈ R∗N , w : E → E restricts to a bijection from the affine

hyperplane Hα,n,N to the affine hyperplane Hwα,n,N . It follows that w : E → E

restricts to a bijection E′ ∼−→ E′.



We show:(b) For any � ∈ E′ we have Ad(g)(εp

�∗ ) = εp

w�∗ .

From 10.2(h) we have

Ad(g)(εp�N ) = ⊕(α,n)∈RN ;〈�:α〉≥2N/η−nAd(g)gα,nN

= ⊕(α,n)∈RN ;〈�:α〉≥2N/η−n(gwα,nN ),

εpw�N = ⊕(α,n)∈RN ;〈w�:α〉≥2N/η−n(g

α,nN )

= ⊕(α′,n)∈RN ;〈w�:wα′〉≥2N/η−n(gwα′,nN ).

It remains to use that 〈� : α〉 = 〈w� : wα〉.

10.12. For �1, �2 in E we set

(a) [�1|�2] = (1− v2)− dimZ∑w∈W

vτ(�2,w�1) ∈ Q(v).

(Here v is an indeterminate and τ (�2, w�1) ∈ Z is as in 10.8.) When �1, �2 arein E′ we have:

(b)∑j∈Z

dj(gδ; I�1, D(I�2

))v−j = [�1|�2].

This can be deduced from [LY, 6.4] as follows. The set X in [LY, 6.3] is describedin our case in 10.11(a) in terms of the group H ′/M0 which, in turn, is identifiedin 10.9(e) with W = H/Z; the integers τ (g) in 6.3 are identified with the integersτ (�2, w�1) ∈ Z by 10.11(b). Finally, the set X ′ in [LY, 6.4] coincides with X in[LY, 6.4] by 10.10(a).

From the definitions we have τ (�2, w�1) = τ (w−1�2, �1) = τ (�1, w−1�2) for

any w ∈ W . It follows that

(c) [�1|�2] = [�2|�1].

Let c1 (resp. c2) be the equivalence class for ∼ in◦E that contains �1 (resp.

�2). Using 10.7(a) we see that the right hand side of (b) depends only on c1, c2

and not on the specific elements �1 ∈ c1 ∩E′, �2 ∈ c2 ∩E′. Hence for c1, c2 in◦E

we can set [c1|c2] = [�1|�2] ∈ Q(v) for any �1 ∈ c1 ∩E′, �2 ∈ c2 ∩E′.

10.13. Let B (resp. ξB) be the set of (isomorphism classes of) simple perversesheaves in Q(gδ) (resp. ξQ(gδ)). For any G0-orbit O in gnilδ let BO be the setof all B ∈ B such that the support of B is equal to the closure of O. We haveB = �OBO. We define a map κ : B → N by κ(B) = dimO where B ∈ BO.

10.14. Let V′ be the Q(v)-vector space with basis {Tc; c ∈◦E}. On V′ we have

a unique pairing (:) : V′ ×V′ → Q(v) which is Q(v)-linear in the first argument,

Q(v)-antilinear in the second argument (for f �→ f) and such that for c1, c2 in◦E

we have (Tc1: Tc2

) = [c1|c2] (see 10.12).Setting

Rl = {x ∈ V′; (x : x′) = 0 ∀x′ ∈ V′},Rr = {x ∈ V′; (x′ : x) = 0 ∀x′ ∈ V′},

we state the following.



Lemma 10.15. We have Rl = Rr.

The proof is given in 10.17.

10.16. We define a Q-linear map

γ : V′ → Q(v)⊗AξK(gδ)

by Tc �→ I�, where � is an element of c∩E′. Now γ is well-defined by 10.7(a) andis surjective by [LY, 8.4(b)], 10.4(b). We define a pairing

(Q(v)⊗A K(gδ))× (Q(v)⊗A K(gδ)) → Q((v))

(denoted by (:)) by requiring that it is Q(v)-linear in the first argument, Q(v)-antilinear in the second argument (for f �→ f) and that its restriction

K(gδ)×K(gδ) → Q((v))

is the same as the restriction of the pairing in [LY, 4.4(c)]. This restricts to apairing

(Q(v)⊗AξK(gδ))× (Q(v)⊗A

ξK(gδ)) → Q((v))

(denoted again by (:)). We show that for b, b′ in V′ we have

(a) (γ(b) : γ(b′)) = (b : b′)

We can assume that b = Tc, b′ = Tc′ with c, c′ in

◦E. We must show that {I�, D(I�′)}

= [�|�′] where � ∈ c ∩ E′, �′ ∈ c′ ∩ E′. This follows from 10.12(b). This proves(a).

10.17. Let

′Rl = {z ∈ Q(v)⊗A

ξK(gδ); (z : z′) = 0 ∀z′ ∈ Q(v)⊗AξK(gδ)},

′Rr = {z ∈ Q(v)⊗A

ξK(gδ); (z′ : z) = 0 ∀z′ ∈ Q(v)⊗A

ξK(gδ)}.We show:(a) ′Rl =

′Rr = 0.Now Q(v)⊗A

ξK(gδ) has a Q(v)-basis formed by ξB = {B1, B2, . . . , Br}. From[LY, 0.12] we see that (Bj : Bj′) ∈ δj,j′ + vN[[v]] for j, j′ in [1, r].

Assume that β =∑

j∈[1,r] fjBj ∈ ′Rl, where fj ∈ Q(v) are not all zero. We

must show that this is a contradiction. We can assume that fj ∈ Q[v] for all j andfj0 − c0 ∈ vQ[v] for some j0 ∈ [1, r] and some c0 ∈ Q− {0}. Then 0 = (β : Bj0) ∈c0 + vQ[[v]], a contradiction. This proves that ′Rl = 0.

Next we assume that β =∑

j∈[1,r] fjBj ∈ ′Rr, where fj ∈ Q(v) are not all

zero. We must show that this is a contradiction. We can assume that fj ∈ Q[v]for all j and fj0 − c0 ∈ vQ[v] for some j0 ∈ [1, r] and some c0 ∈ Q − {0}. Then0 = (Bj0 : β) ∈ c0+vQ[[v]], a contradiction. This proves that ′Rr = 0. This proves(a).

We show:(b) Rl = γ−1(′Rl).Let x ∈ Rl. From 10.16(a) we see that (γ(x) : γ(x′)) = 0 for any x′ ∈ V′. Since

γ is surjective, it follows that (γ(x) : z′) = 0 for any z′ ∈ Q(v) ⊗AξK(gδ). Thus,

γ(x) ∈ ′Rl. Conversely, assume that x ∈ V′ and γ(x) ∈ ′Rl. From 10.16(a) we seethat for any x′ ∈ V′ we have (x : x′) = (γ(x) : γ(x′)) = 0. Thus x ∈ Rl. Thisproves (b).



An entirely similar proof shows that:(c) Rr = γ−1(′Rr).From (a),(b),(c) we see that Rl = Rr = γ−1(0). This proves Lemma 10.15.

10.18. Definition of V. We define V = V′/Rl = V′/Rr (see Lemma 10.15).Note that (:) on V′ induces a pairing V×V → Q(v) (denoted again by (:)) whichis Q(v)-linear in the first argument, Q(v)-antilinear in the second argument (forf �→ f).

10.19. From the proof of Lemma 10.15 we see that γ induces a Q(v)-linear isomor-phism

(a) γ : V∼−→ Q(v)⊗A

ξK(gδ)

and that for b, b′ in V we have

(b) (γ(b) : γ(b′)) = (b : b′).

From (a) we deduce the following result.

Proposition 10.20. The number of simple perverse sheaves (up to isomorphism)in ξQ(gδ) is equal to dimQ(v) V.

10.21. We define a Q-linear involution¯: V′ → V′ by fTc = f Tc for any f ∈ Q(v),

c ∈◦E; here f is as in [LY, 0.12]. We show:

(a) For any x, x′ in V′ we have (x : x′) = (x′ : x).

We can assume that x = Tc, x′ = Tc′ for some c, c′ in

◦E. We must show that

[c|c′] = [c′|c],which follows directly from 10.12(c). This proves (a).

We show:(b) ¯ : V′ → V′ preserves Rl = Rr (see Lemma 10.15) hence it induces an

Q(v)-semilinear involution¯: V → V (with respect to f �→ f).Assume that x ∈ Rl. We have (x : x′) = 0 for all x′ ∈ V. Hence, by (a), we have

(x′ : x) = 0 for all x′ ∈ V. Since¯: V′ → V′ is surjective, it follows that x ∈ Rr.This proves (b).

10.22. Now let η1 ∈ Z − {0} be such that η1= δ. Recall that in 10.1 we have

fixed ξ ∈ Tη and a representative ξ = (M,M0,m,m∗, C) ∈ Tη for ξ. Let ξ1 =

(M,M0,m,m(∗), C) ∈ Tη1be as in [LY, 3.9] and let ξ1 ∈ Tη1

be the G0-orbit of x1.

Note that the Q-vector space E defined as in 10.1 in terms of ξ is the same as E

defined in terms of ξ1. Moreover, the subset◦E, the equivalence relation ∼ on it,

and the set◦E defined as in 10.7 in terms of ξ are the same as the analogous objects

defined in terms of ξ1. Also, the pairings τ : E×E → Z and [?|?] : E×E → Q(v)

defined in 10.8 and 10.12 in terms of ξ are the same as those defined in terms of ξ1.The subset E′ of E defined as in 10.1 in terms of ξ is not in general the same as

the analogous subset E′1 of E defined in terms of ξ1. However, for c1, c2 in

◦E, the

quantity [c1|c2] ∈ Q(v) defined in 10.12 in terms of ξ is the same as that defined in

terms of ξ1. (It is equal to [�1|�2] for any �1 ∈ c1 ∩E′ ∩ E′1, �2 ∈ c2 ∩E′ ∩ E′

1.Hence the vector space V′, the pairing (:) on it and its quotient V defined in 10.14



and 10.18 in terms of ξ are the same as those defined in terms of ξ1. The involutions¯: V′ → V′,¯: V → V defined in 10.21 in terms of ξ are the same as those definedin terms of ξ1.

11. The A-lattice VA

In this section we give a combinatorial definition of an A-lattice VA in the Q(v)-vector space V and a signed basis B′ of it. It turns out that the Z/2-orbits on B′

for the Z/2-action b �→ −b are in natural bijection with the simple perverse sheavesin the block ξQ(gδ).

11.1. In this section (except in 11.5, 11.6, 11.13) we preserve the setup and notationof 10.1, 10.2 and assume that � ∈ E′′ (see 10.3). Let

gφ = {y ∈ g; [y, e] = 0, [y, h] = 0, [y, f ] = 0}.

Let z be the center of m. Note that m ∩ gφ = z (since e is distinguished in m) andz is a Cartan subalgebra of the reductive Lie algebra gφ. (This has already beenproved at level of groups in [LY, 3.6].) For any α ∈ XZ let

(gφ0 )α = gα0 ∩ gφ.

Let

gφ0,� = ⊕α∈XZ ;〈�:α〉=0(g

φ0 )

α.

This is a Levi subalgebra (containing z) of a parabolic subalgebra of gφ0 . Let B be

the variety of Borel subalgebras of gφ0,�, let d(�) = dimB and let

a� =∑j

v−2sjvd(�) ∈ A,

where ρB!Ql = ⊕jQl[−2sj ].Erratum to [L4]. On page 202, line 1 of 16.8, replace “algebraic group M” by

“algebraic group M with a given Lie algebra homomorphism φ from s to the Liealgebra of M”.

On page 202, line 4 of 16.8, replace “Borel subgroups of M” by “Borel subgroupsof the connected centralizer of φ(s) in M”.

11.2. The subset

(a) {a−1� Tc ∈ V′; c ∈

◦E, � ∈ E′′ ∩ c}

of V′ (see 10.14) is finite. Indeed, when c is fixed, the subgroup gφ0,� of gφ0 takes

only finitely many values for � in E′′ ∩ c hence a� takes only finitely many values.Let V′

A be the A-submodule of V′ generated by (a). Let VA be the image ofV′

A under the obvious linear map V′ → V. The following result will be proved in11.8.

(b) VA is a free A-module such that the obvious Q(v)-linear map Q(v)⊗AVA →V is an isomorphism.



11.3. Let h = εl�, hφ = h ∩ gφ. We show:

(a) gφ0,� = hφ.

From 10.3(b) we have

h = ⊕N∈Z,(α,n)∈RN ;n=2N/η,〈�:α〉=0(gα,nN ).

Using this and the definitions we have

{y ∈ h; [y, h] = 0} = ⊕N∈Z,(α,n)∈RN ;n=2N/η=0,〈�:α〉=0(gα,nN )

= ⊕(α,n)∈R0;n=0,〈�:α〉=0(gα,n0 )

and (a) follows.

11.4. Recall that � ∈ E′′. Let p∗ = εp�∗ , u∗ = εu

�∗ . We can find λ′ ∈ YZ such

that 〈λ′ : α〉 �= 0 for any i and any (α, n) ∈ R∗i . Let �′ = � + 1

bλ′ ∈ E. Let

p′∗ = εp�′

∗ , u′∗ = εu�′

∗ and m′∗ = εl

�′

∗ . Assume that b is sufficiently large; then�′ ∈ E′ hence by 10.2 we have m′

∗ = m∗. The same argument as in 10.6(a) showsthat p′N ⊂ pN for all N . Since b is large and �′ = � + 1

bλ′, we see that �′, � are

very close, so that

(a) �′ ∼ �.

For N ∈ Z let qN be the image of p′N under the obvious projection pN → hN . From10.5(a),(b), we see that q = ⊕NqN is a parabolic subalgebra of h and m is a Levisubalgebra of q. Moreover, if uN is the image of u′N under pN , then u = ⊕NuN isthe nilradical of q.

From 10.3(b) we see that the Z-grading of h is η-rigid and that e ∈◦hη, so that

◦mη ⊂

◦hη. Let A� ∈ Q(hη) be the simple perverse sheaf on hη such that the support

of A� is hη and A�| ◦mη

is equal up to shift to C| ◦mη

.

Applying 1.8(b) and the transitivity formula 4.2(a) we deduce

(b)

I�′ = εIndgδ

p′η(C) = εInd

gδ

pη(indhη

qη(C))[dim u

′0 + dim u

′η − dim u0 − dim uη]

∼= ⊕jεInd

gδ

pη(A�)[−2sj ][dimmη − dim hη + dim u′0 + dim u′η

− dim u0 − dim uη]

= ⊕jεInd

gδ

pη(A�)[−2sj ][d(�)],

where we have used the equality:

(c) dimmη − dim hη + dim u′0 + dim u

′η − dim u0 − dim uη = d(�).

We now prove (c). Let u′ be the nilradical of the parabolic subalgebra of h thatcontains m and is opposed to q. We have u′ = ⊕Nu′

N where u′N = u′ ∩ hN . Now

u0, u′0 are nilradicals of two opposite parabolic subalgebras of h0 hence dimu0 =

dimu′0. We have dim hη = dimuη + dim u′

η + dimmη, dim u′0 − dim u0 = dimu0,dim u′η −dim uη = dim uη. Hence the left-hand side of (c) equals dim u0−dim u′

η =dimu′

0 − dimu′η. Now u′ is normalized by m hence by the Lie subalgebra s of

m spanned by e, h, f . Note that u′0 (resp. u′

η) is the 0-(resp. 2-) eigenspace ofad(h) : u′ → u′. By the representation theory of s, the map ad(e) : u′

0 → u′η is

surjective and its kernel is exactly the space of s-invariants in u′ that is u′ ∩ hφ.We see that dimu′

0 − dimu′η = dim(u′ ∩ hφ). Now u′ ∩ hφ is the nilradical of a



parabolic subalgebra of hφ with Levi subgroup m ∩ hφ = z (see 11.1) hence is the

nilradical of a Borel subalgebra of hφ (which equals gφ0,� by 11.3(a)). By definition,

the dimension of this nilradical is equal to d(�). This proves (c) and hence also(b).

Now (b) implies the following equality in ξK(gδ):

I�′ = a�(εIndgδ

pη(A�)).

Using this and (a) we see that for any c ∈◦E and any � ∈ E′′ ∩ c we have

(d) a−1� γ(Tc) =

εIndgδ

pη(A�) ∈ Q(v)⊗A

ξK(gδ).

Since εIndgδ

pη(A�) ∈ ξK(gδ), we see that:

(e) For any c ∈◦E and any � ∈ E′′ ∩ c we have a−1

� γ(Tc) ∈ ξK(gδ).The following result will be proved in 11.7.

(f) The A-module ξK(gδ) is generated by the elements a−1� γ(Tc) for various c ∈

◦E

and � ∈ E′′ ∩ c.From (f) we deduce:

(g) The isomorphism γ : V∼−→ Q(v)⊗A

ξK(gδ) in Proposition 10.19(a) restricts

to an isomorphism of A-modules γA : VA∼−→ ξK(gδ).

11.5. Let p∗ be an ε-spiral with a splitting h∗ and with nilradical u∗; let A ∈ Q(hη)be a simple perverse sheaf. We show:

(a) Let X be the collection of all B ∈ B such that some shift of B is a direct

summand of Indgδ

pη(A). Then the map X → Tη, B �→ ψ(B) is constant.

We can find a parabolic subalgebra q of h and a Levi subalgebra m′ of q such thatq = ⊕NqN ,m′ = ⊕Nm′

N where qN = q∩hN , m′N = m′∩hN and a cuspidal perverse

sheaf C in Q(mη) such that some shift of A is a direct summand of indhηqη(C).

Let M ′ = em′,M ′

0 = em′0 . Setting p′N = uN + qN for any N ∈ Z, we see from

[LY, 2.8(a)] that p′∗ is an ε-spiral and from [LY, 2.8(b)] that m′∗ is a splitting of

p′∗. We see that (M ′,M ′0,m

′,m′∗, C) ∈ Tη. Let ξ

′ be the element of Tη determinedby (M ′,M ′

0,m′,m′

∗, C). If B ∈ X , then (by [LY, 4.2]) some shift of B is a direct

summand of Indgδ

p′η(C) hence ψ(B) = ξ′. This proves (a).

We say that Indgδ

pη(A) (as in (a)) has type ξ′ ∈ Tη if ψ(B) = ξ′ for any B ∈ X .

11.6. Recall from 8.4(a) that the A-module K(gδ) is generated by the classes ofε-quasi-monomial objects of Q(gδ). Using this and 11.5(a), we deduce that in thedirect sum decomposition K(gδ) = ⊕ξ∈Tη

ξK(gδ) (see [LY, 6.7]), any summandξK(gδ) is generated as an A-module by the classes of η-quasi-monomial objects inQ(gδ) of type ξ.

11.7. We prove 11.4(f). (Thus we are again in the setup of 11.1.) Using 11.6, wesee that it is enough to show that if A′ is an η-quasi-monomial object in Q(gδ) of

type ξ, then the class of A′ in K(gδ) is of the form a−1� γ(Tc) for some c ∈

◦E and

� ∈ E′′ ∩ c. We can find:(a) p∗, h∗, A, q∗, p

′∗, (M ′,M ′

0,m′,m′

∗, C) ∈ Tη (representing ξ) as in 11.5 such

that A′ = Indgδ

pη(A); moreover, we can assume that the Z-grading of h = ⊕NhN is



η-rigid and◦m′

η ⊂◦hη. Replacing the data (a) by a G0-conjugate we can assume in

addition that (M ′,M ′0,m

′,m′∗, C) is equal to (M,M0,m,m∗, C) in 10.1.

Let H = eh. Since h∗ is η-rigid, there exists ι′ ∈ YH such that:(i) ι′

k = hkη/2 if k ∈ Z, kη/2, ι′

k = 0 if k ∈ Z, kη/2 /∈ Z and

(ii) ι′ = ιφ′ for some φ′ = (e′, h′, f ′) ∈ JH such that e′ ∈◦hη, h

′ ∈ h0, f′ ∈ h−η.

Since◦mη ⊂

◦hη and e ∈ ◦

mη, we see that e, e′ are in the same M0-orbit. Hence wecan find g ∈ M0 such that Ad(g) conjugates e′, f ′, h′, ι′ to e, f, h, ι. Applying Ad(g)(which preserves hk) to (i) and (ii) we see that we can assume that ι′ = ι, φ′ = φ.

Recall from 10.3 that lφN = ι

2N/ηgN for N ∈ Z such that 2N/η ∈ Z. Using

(i) with ι′ = ι we see that hN ⊂ lφN for any N ∈ Z such that 2N/η ∈ Z. Using

10.4(c),(d) we see that for some � ∈ E′′ we have p∗ = εp�∗ , h∗ = εl

�∗ . Using now

11.4(d), we see that A′ = a−1� γ(Tc), where c ∈

◦E contains �. This completes the

proof of 11.4(f), hence that of 11.4(g).

11.8. We can now prove 11.2(b). Using 11.4(g), 11.2(b) is reduced to the followingobvious statement: ξK(gδ) is a free A-module.

11.9. We define a Q-linear map

¯: Q(v)⊗A K(gδ) → Q(v)⊗A K(gδ)

by fB = fB for any f ∈ Q(v) and any B ∈ B (see 10.13); here f is as in 0.12.This restricts to a Q-linear map

¯: Q(v)⊗AξK(gδ) → Q(v)⊗A

ξK(gδ)

and to a Z-linear map ξK(gδ) → ξK(gδ). We show:(a) I� = I� for any � ∈ E′.In ξQ(gδ) we have I� =

∑B∈ξB

fBB where fB ∈ A. It is enough to prove that

fB = fB for all B.We set p∗ = εp

�∗ . Let σ be an automorphism of order 2 of Ql such that σ(z) =

z−1 for any root of 1 in Ql. Applying σ to K ∈ Q(mη) (resp. K ∈ Q(gδ)) weobtain Kσ ∈ Q(mη) (resp. Kσ ∈ Q(gδ)). Note that K �→ Kσ commutes withshifts; moreover, we have

(b) (εIndgδ

pη(C))σ = εInd

gδ

pη(Cσ).

Moreover, if K is a simple perverse sheaf in Q(mη) or in Q(gδ), we have

(c) Kσ ∼= D(K),

since K restricted to an open dense subset of it support is a local system with finitemonodromy. By (b),(c) and [LY, 4.1(d)] we have

D(εIndgδ

pη(C)) = εInd

gδ

pη(D(C)) = εInd

gδ

pη(Cσ) = (εInd

gδ

pη(C))σ,

hence

(D(I�))σ = I�

and∑

B fBD(Bσ) =∑

B fBB. Using this and (c) for K = B, we see that fB = fBfor all B; this proves (a).



11.10. We show:(a)¯: V′ → V′ (see 10.21) restricts to an involution V′

A → V′A and¯: V → V

(see 10.21) restricts to an involution VA → VA (these restrictions are denotedagain by .

It is enough to note that for any � ∈ E′′ we have a� = a�.

We show:(b) γ : V′ → Q(v)⊗A

ξK(gδ) is compatible with the maps¯ in the two sides.This follows from 11.9(a).

We have the following result.

Proposition 11.11.(a) Let B′ be the set of all b ∈ VA such that b = b and (b : b) ∈ 1+vZ[[v]]. Then

B′ is a signed basis of the A-module VA (that is, the union of a basis with (−1)times that basis).

(b) For b ∈ B′ we have (b : b) ∈ 1 + vN[[v]].

(c) There is a unique A-basis B of VA such that B ⊂ B′ and for any c ∈◦E and

any � ∈ E′′ ∩ c, the image of a−1� Tc in VA is an N[v, v−1]-linear combination of

elements in B.

It is enough to prove the analogous statements where VA is identified via γ withξK(gδ) with (:) as in 4.4(c) and with¯as in 11.9 (we use 10.19(b), 11.10(b)). LetξB = {B1, B2, . . . , Br} (see 10.13). From 0.12 we have (Bj : Bj′) ∈ δj,j′ + hj,j′ ,where hj,j′ ∈ vN[[v]] for all j, j′ in [1, r]. From the definition (see 11.9) we have

Bj = Bj for j = 1, . . . , r. Now let b ∈ ξK(gδ) be such that b = b and (b : b) ∈1+vZ[[v]]. To prove (a), it is enough to show that b = ±Bj for some j. We can writeb =

∑rj=1 fjBj , where fj ∈ A satisfy fj = fj and

∑j,j′∈[1,r] fjfj′(δj,j′ + hj,j′) ∈

1 + vZ[[v]] hence∑

j,j′∈[1,r] fjfj′(δj,j′ + hj,j′) ∈ 1 + vZ[[v]]. We can find c ∈ Z

such that fj = fj,cvc mod vc+1Z[v] where fj,c ∈ Z for all j and fj,c �= 0 for some

j. We have∑

j∈[1,r] f2j,cv

2c + v2c+1f ′ = 1 + vf ′′ where f ′, f ′′ ∈ Z[v]. Moreover,∑j∈[1,r] f

2j,c > 0. It follows that c = 0 and

∑j∈[1,r] f

2j,0 = 1 so that there exists

j0 ∈ [1, r] such that fj0,0 = ±1 and fj,0 = 0 for j �= j0. We have fj = ±δj,j0mod vZ[v] for all j. Since fj = fj we deduce that fj = ±δj,j0 for all j. Thusb = ±Bj0 . This completes the proof of (a). At the same time we have proved (b).Clearly, {B1, B2, . . . , Br} has the positivity property in (c) (with VA identified

with ξK(gδ) and with a−1� Tc identified with γ(a−1

� Tc)). Since any Bj appears

with > 0 coefficient in some γ(a−1� Tc), we see that {B1, B2, . . . , Br} is the only

basis contained in {±B1,±B2, . . . ,±Br} with the positivity property in (c). Thiscompletes the proof of the proposition.

11.12. From the proof of 11.11 we see that γ : V∼−→ Q(v)⊗A

ξK(gδ) (see Proposition10.19(a)) restricts to a bijection

(a) B∼−→ ξ

B.

For any G0-orbit O in gnilδ let BO be the set of all b ∈ B such that γ(b) ∈ BO (see

10.13). We have a partition B = �OBO where O runs over the G0-orbits in gnilδ .



11.13. We consider the setup of 10.22. We show:(a) The A-submodule VA of V defined in 11.2 in terms of ξ is the same as that

defined in terms of ξ1.It is not clear how to prove this using the definition in 11.2 since E′′ defined

in terms of ξ (see 10.3) is not necessarily the same as that defined in terms of ξ1.Instead we will argue indirectly. Using 11.4(g) it is enough to show:

(b) The isomorphism γ : V∼−→ Q(v)⊗A

ξK(gδ) defined in 10.19 in terms of ξ is

equal to the analogous isomorphism defined in terms of ξ1.Thus it is enough to show that if � ∈ E′ ∩E′

1, then I� defined in 10.2 in terms

of ξ is the same as that defined in terms of ξ1. Using the definitions we see that itis enough to show that

ηp(|η|/2)(�+ι)η = η1p(|η1|/2)(�+ι)

η1,

ηp(|η|/2)(�+ι)0 = η1p

(|η1|/2)(�+ι)0 .

or that⊕κ∈Q;κ≥|η|(

(|η|/2)(�+ι)κ gδ) = ⊕κ∈Q;κ≥|η1|(

(|η1|/2)(�+ι)κ gδ),

⊕κ∈Q;κ≥0((|η|/2)(�+ι)κ gδ) = ⊕κ∈Q;κ≥0(

(|η1|/2)(�+ι)κ gδ),

or that⊕κ∈Q;κ/|η|≥1(

(1/2)(�+ι)κ/|η| gδ) = ⊕κ∈Q;κ/|η1|≥1(

(1/2)(�+ι)κ/|η1| gδ),

⊕κ∈Q;κ/|η|≥0((1/2)(�+ι)κ/|η| gδ) = ⊕κ∈Q;κ/|η1|≥0(

(1/2)(�+ι)κ/|η1| gδ),

or, setting κ′ = κ/|η|, κ′′ = κ/|η1|, that

⊕κ′∈Q;κ′≥1((1/2)(�+ι)κ′ gδ) = ⊕κ′′∈Q;κ′′≥1(

(1/2)(�+ι)κ′′ gδ),

⊕κ′∈Q;κ′≥0((1/2)(�+ι)κ′ gδ) = ⊕κ′′∈Q;κ′′≥0(

(1/2)(�+ι)κ′′ gδ),

which are obvious. This proves (a).

Using (b) and 11.12(b) we see that the basis B of VA defined in 11.11 in terms

of ξ is the same as that defined in terms of ξ1.

12. Purity properties

In this section we show that for any irreducible local system L on a G0-orbit in

gnilδ the cohomology sheaves of L� ∈ D(gδ) satisfy a strong purity property. Thisgeneralizes the analogous result in the Z-graded case in [L4].

12.1. In this section we assume that p > 0 and that k is an algebraic closure of afinite field Fq with q elements (here q is a power of p). Replacing q by larger powersof p if necessary, we can assume that m divides q − 1 and that we can find an Fq-rational structure on G with Frobenius map F : G → G such that ϑ : G → G (see0.5) commutes with F : G → G. Then G0 is defined over Fq and g inherits fromG an Fq-rational structure with Frobenius map F : g → g satisfying F (gi) = gi forall i. Again by replacing q by larger powers of p if necessary, we may assume thatall G0-orbits in gnilδ are defined over Fq and that for any irreducible G0-equivariant

local system L on a G0-orbit O in gnilδ we have F ∗L ∼= L.We now fix a G0-orbit O in gnilδ with closure O and an irreducible G0-equivariant

local system L on O. We fix an isomorphism F : F ∗L → L which induces theidentity map on the stalk of L at some point ofOF . Then F induces an isomorphism(denoted again by F ) F ∗L� → L�. Given a finite-dimensional Ql-vector space V



with an endomorphism F : V → V , we say that F : V → V is a-pure (for an integer

a) if the eigenvalues of F are algebraic numbers all of whose complex conjugateshave absolute value qa/2. Sometimes we will just say that V is a-pure (where this

is understood to refer to F ).

12.2. We show:(a) For any x ∈ OF and any a ∈ Z, the induced linear map F : Ha

x(L�) → Hax(L�)

is a-pure.Using [LY, 2.3(b)], we can find φ = (e, h, f) ∈ Jδ(x) such that h, f are Fq-

rational. Recall that x = e. Let ι = ιφ ∈ YG be as in 1.1. Let z(f) be thecentralizer of f in g and let Σ = e+ z(f) ⊂ g. According to Slodowy:

(b) The affine space Σ is a transversal slice at x to the G-orbit of e in g andthe k∗-action t �→ t−2 Ad(ι(t)) on g keeps e fixed, leaves Σ stable and defines acontraction of Σ to x.

Let Σ = Σ ∩ gδ = e+ (z(f) ∩ gδ). Then:

(c) Σ is a transversal slice to the G0-orbit of e in gδ; the k∗-action in (b) leaves

stable Σ and is a contraction of Σ to e.(We have a direct sum decomposition gδ = (z(f)∩gδ)⊕ [e, g0] obtained by taking

the ζδ-eigenspace of θ : g → g in the two sides of the direct sum decompositiong = z(f)⊕ [e, g]; note that both z(f) and [e, g] are θ-stable.)

Let L′ be the restriction of L to O ∩ Σ (a smooth irreducible subvariety of Σ).

Note that O∩Σ, O∩Σ are stable under the k∗-action in (c) and L′ is equivariant forthat action. By (c), we have Hx(L�) = Hx(L′�). It remains to show that Hx(L′�) isa-pure. This can be reduced to Deligne’s hard Lefschetz theorem by an argumentin Lemma 4.5(b) in [KL2] applied to O ∩ Σ ⊂ Σ with the contraction in (c) and toL′ instead of Ql. Note that in [KL2, 4.5(b)] an inductive purity assumption wasmade which is in fact unnecessary, by Gabber purity theorem. This completes theproof of (a).

Erratum to [L4]. On page 209, line -5, replace t−n Ad(ι′(t)) by t−2 Ad(ι′(t)).

13. An inner product

This section is an adaptation of [L4, §19] from the Z-graded case to the Z/m-graded case; we express the matrix whose entries are the values of the (:)-pairing attwo elements of B as a product of three matrices. As an application we show (see13.8(a)) that if (O,L), (O′,L′) in I(gδ) are such that some cohomology sheaf ofL�|O′ contains L′, then (O,L), (O′,L′) are in the same block. Another application(to odd vanishing) is given in Section 14.

13.1. We fix (O,L), (O′,L′) in I(gδ) and we form A = L�, A′ = L′� in D(gδ). Wewant to compute dj(gδ;A,A′) (see [LY, 0.12]) for a fixed j ∈ Z. We can arrangethe G0-orbits in gnilδ in an order O0,O1, . . . ,Oβ such that O≤s = O0∪O1∪· · ·∪Os

is closed in gδ for s = 0, 1, . . . , β. We choose a smooth irreducible variety Γ witha free action of G0 such that Hr(Γ, Ql) = 0 for r = 1, 2, . . . ,m where m is a largeinteger (compared to j). We assume that D := dimΓ is large (compared to j). We

have H2D−ic (Γ, Ql) = 0 for i = 1, 2, . . . ,m. We form X = G0\(Γ×gδ). Let L, L′ be

the local systems on the smooth subvarieties G0\(Γ×O), G0\(Γ×O′) of X defined

by L,L′ and let A = L�, A′ = L′� be the corresponding intersection cohomologycomplexes in D(X). Then A⊗ A′ ∈ D(X) is well defined; its restriction to various



subvarieties of X will be denoted by the same symbol. For s = 0, 1, . . . , β we formXs = G0\(Γ×Os), X≤s = G0\(Γ×O≤s). We set X≤−1 = ∅. The partition of X≤s

into X≤s−1 and Xs (for s = 0, 1, . . . , β) gives rise to a long exact sequence

(a). . .

ξa−→ Hac (Xs, A⊗ A′) → Ha

c (X≤s, A⊗ A′)

→ Hac (X≤s−1, A⊗ A′)

ξa+1−−−→ Ha+1c (Xs, A⊗ A′) → . . . .

13.2. In the following proposition we encounter two kinds of integers; some like j,m, dimG0, dim gδ, β are regarded as “small” (they belong to a fixed finite set ofintegers), others like 2D and m are regarded as very large (we are free to choosethem so). We will also encounter integers a such that 2D − a is a “small” integer(we then write a ∼ 2D).

Proposition 13.3.(a) Assume that k is as in 12.1. Then Ha

c (Xs, A ⊗ A′) is a-pure (see 12.1) fors = 0, 1, . . . , β and a ∼ 2D.

(b) We choose xs ∈ Os. If a ∼ 2D, then

dimHac (Xs, A⊗ A′)

=∑

r+r1+r2=a

dim(Hrc (G0(xs)

0\Γ, Ql)⊗Hr1xsA⊗Hr2

xsA′)G0(xs)/G0(xs)

0

,

where the upper script refers to invariants under the finite group G0(xs)/G0(xs)0.

(c) Assume that k is as in 12.1. Then Hac (X≤s, A⊗ A′) is a-pure (see 12.1) for

s = 0, 1, . . . , β and a ∼ 2D.(d) The exact sequence 13.1(a) gives rise to short exact sequences

0 → Hac (Xs, A⊗ A′) → Ha

c (X≤s, A⊗ A′) → Hac (X≤s−1, A⊗ A′) → 0

for s = 0, 1, . . . , β and a ∼ 2D.

(e) For a ∼ 2D we have dimHac (X, A⊗ A′) =

∑βs=0 dimHa

c (Xs, A⊗ A′).(f) For a ∼ 2D we have

dimHac (X, A⊗ A′)

=

β∑s=0

∑r+r1+r2=a

dim(Hrc (G0(xs)

0\Γ, Ql)⊗Hr1xsA⊗Hr2


0

.

The proof is almost a copy of the proof of [L4, 19.4]. By general principles wecan assume that k is as in 12.1. We shall use Deligne’s theory of weights. Wefirst prove (a) and (b). We write x instead of xs. We may assume that x is anFq-rational point. We have a natural spectral sequence

(g) Er,r′

2 = Hrc (Xs,Hr′(A⊗ A′)) =⇒ Hr+r′

c (Xs, A⊗ A′).

We show that

(h) Er,r′

2 is (r + r′)-pure if r + r′ ∼ 2D.

We have Xs = G0(x)\Γ and

Er,r′

2 = (Hrc (G0(x)

0\Γ, Ql)⊗Hr′

x (A⊗A′))G0(x)/G0(x)0

.



Here A ⊗ A′ ∈ D(gδ). We may assume that r′ is “small” (otherwise, Er,r′

2 = 0).We then have r ∼ 2D. Now

Hr′

x (A⊗A′) = ⊕r1+r2=r′Hr1x (A)⊗Hr2

x (A′)

is r′-pure by 12.2(a). Moreover, Hrc (G0(x)

0\Γ, Ql) is r-pure for r ∼ 2D (a knownproperty of the classifying space of G0(x)

0) and (h) follows.

From (h) it follows that Er,r′

∞ of the spectral sequence (g) is (r + r′)-pure if

r+r′ ∼ 2D and (a) follows. From (h) it also follows that Er,r′

2 = Er,r′

∞ if r+r′ ∼ 2D(many differentials must be zero since they respect weights) and (b) follows.

Now (c) follows from (a) using the exact sequence 13.2(a) and induction on s.From (c) and (a) we see that the homomorphism ξa+1 in 13.2(a) is between purespaces of different weight. Since ξa+1 preserve weights, it must be 0. Similarly,ξa = 0 in 13.2(a) hence (d) holds. Now (e) follows from (d) since the support of

A ⊗ A′ is contained in X≤β; (f) follows from (b),(e). This completes the proof ofthe proposition.

13.4. Using the definitions we see that 13.3(f) implies:(a)

dj(gδ;A,A′) =∑s

∑−r0+r1+r2=j−2ps

dim(HG0(xs)

0

r0 (pt)⊗Hr1xsA⊗Hr2


0

,

where

(b) ps = dimG0 − dimG0(xs) = dimOs

and HG0(xs)

0

r0 (pt) denotes equivariant homology of a point. (See [L6] for the defini-tion of equivariant homology.)

13.5. Given (O,L), (O, L) in I(gδ) we define μ(L,L) ∈ Z[v−1] by

(a) μ(L,L) =∑a

μ(a; L,L)v−a,

where μ(a; L,L) is the number of times L appears in a decomposition of the local

system Ha(L�)|O as a direct sum of irreducible local systems. Note that μ(L,L) iszero unless O is contained in the closure of O.

If E is an irreducible G0-equivariant local system on Os, we denote by ρE theirreducible G0(xs)/G0(xs)

0-module corresponding to E . With this notation we canrewrite 13.4(a) as follows:

dj(gδ;A,A′) =∑s

∑−r0+r1+r2=j−2ps∑

E,E′

μ(r1; E ,L)μ(r2; E ′,L′) dim(HG0(xs)

0

r0 (pt)⊗ ρE ⊗ ρE′)G0(xs)/G0(xs)0

,

where E , E ′ run over the isomorphism classes of irreducible G0-equivariant localsystems on Os. This may be written in terms of power series in Q((v)) as follows:

(b) {A,A′} =

β∑s=0

∑E,E′

μ(E ,L)Ξ(E , E ′)μ(E ′,L′),



where

(c) Ξ(E , E ′) =∑r0≥0

dim(HG0(xs)

0

r0 (pt)⊗ ρE ⊗ ρE′)G0(xs)/G0(xs)0

vr0−2ps ∈ Q((v)).

13.6. Let B1, B2 ∈ B. We write B1 = L�1[dimO1] ∈ D(gδ), B2 = L�

2[dimO2] ∈D(gδ) with (O1,L1), (O2,L2) in I(gδ). We set

PB2,B1= μ(L2,L1) ∈ N[v−1], (see 13.5);

PB2,B1= PD(B2),D(B1) ∈ N[v−1];

ΛB2,B1= Ξ(L2,L1) ∈ Q((v)), (see 13.5) if O1 = O2;

ΛB2,B1= 0 if O1 �= O2;

ΛB2,B1= ΛB2,D(B1) ∈ Q((v)).

Note that:(a) PB2,B1

�= 0 =⇒ dimO2 ≤ dimO1; PB2,B1�= 0 =⇒ dimO2 ≤ dimO1;

(b) if dimO2 = dimO1, then PB2,B1= δB2,B1

, PB2,B1= δB2,B1

;

(c) ΛB2,B1= 0 if O1 �= O2.

Then 13.5(b) can be rewritten as follows:

{A,A′} =∑

B1∈B,B2∈B

PB1,BΛB1,B2PB2,B′′ ,

where B = A[dimO], B′′ = A′[dimO′], or as

{A,A′} =∑

B1∈B,B2∈B

PB1,BΛB1,D(B2)PD(B2),D(B′),

where B = A[dimO], D(B′) = A′[dimO′], or as

{A,A′} =∑

B1∈B,B2∈B

PB1,BΛB1,B2PB2,B′ ,

where B = A[dimO], D(B′) = A′[dimO′]. We have

{A,A′} = v−κ(B)−κ(B′){B,D(B′)} = v−κ(B)−κ(B′)(B : B′),

hence

(d) v−κ(B)−κ(B′)(B : B′) =∑

B1∈B,B2∈B

PB1,BΛB1,B2PB2,B′

for any B,B′ in B. (Here κ is as in 10.13.) We show:(e) The following three matrices with entries in Q((v)) (indexed by B×B) are

invertible:(i) the matrix ((B : B′));

(ii) the matrix M := (v−κ(B)−κ(B′)(B : B′));

(iii) the matrix T := (ΛB,B′).The matrix in (i) is invertible since (B : B′) ∈ δB,B′ + vN[[v]] for all B,B′; see

[LY, 0.12]. This implies immediately that M is invertible. Now, by (d), we haveM = ST S ′ where S (resp. S ′) is the matrix indexed by B×B whose (B,B′)-entry

is PB′,B (resp. PB,B′). Since M is invertible, it follows that T is invertible. Thisproves (e).



13.7. We show:(a) If B,B′ in B satisfy ψ(B) �= ψ(B′) (ψ as in [LY, 6.6]), then PB,B′ = 0,

PB,B′ = 0, ΛB,B′ = 0.We can find a function B → Z, B �→ cB such that for B,B′ in B we have

cB = cB′ if and only if ψ(B) = ψ(B′). From 13.6 we deduce

vcB−cB′ v−κ(B)−κ(B′)(B : B′)

=∑

B1∈B,B2∈B

vcB−cB1PB1,BvcB1

−cB2 ΛB1,B2vcB2

−cB′ PB2,B′ .

When B,B′ vary, this again can be expressed as the decomposition of the matrixM := (vcB−cB′ v−κ(B)−κ(B′)(B : B′)) (indexed by B × B) as a product of three

matrices ST S ′, where S (resp. S ′) is the matrix indexed by B×B whose (B,B′)-

entry is vcB−cB′PB′,B (resp. vcB−cB′ PB,B′) and T is the matrix indexed by B×B

whose (B,B′)-entry is vcB−cB′ ΛB,B′ . From [LY, 7.9(a)] we know that (B : B′) = 0unless ψ(B) = ψ(B′). Hence vcB−cB′ (B : B′) = (B : B′) for all B,B′ so that

M = M. Thus we have

ST S ′ = ST S ′.

Now by 13.6(c),(e), the matrix T (and hence the matrix T ) belongs to a subgroupof GLN (N = �(B)) of the form GLN1

× · · · × GLNkwhere N1, . . . , Nk are the

sizes of the various subsets BO; moreover, by 13.6(a),(b), the matrix S (hence the

matrix S) belongs to the unipotent radical of a parabolic subgroup of GLN withLevi subgroup equal to the subgroup of GLN considered above, while the matrixS ′ (hence the matrix S ′) belongs to the unipotent radical of the opposite parabolic

subgroup with the same Levi subgroup. This forces the equalities S = S, T = T ,S ′ = S ′. Now the equality S = S implies vcB−cB′PB′,B = PB′,B for all B′, B inB. Thus, if ψ(B) �= ψ(B′) (so that cB �= cB′), we must have PB′,B = 0. Similarly,

from T = T we see that, if ψ(B) �= ψ(B′), then ΛB,B′ = 0 and from S ′ = S ′ we

see that, if ψ(B) �= ψ(B′), then PB,B′ = 0. This proves (a).

13.8. We now fix ξ ∈ Tη. We define four matrices ξM, ξS, ξT , ξS ′ indexed byξB× ξB as follows. For B,B′ in ξB, the (B,B′)-entry of ξM is v−κ(B)−κ(B′)(B :

B′); the (B,B′)-entry of ξS is PB′,B ; the (B,B′)-entry of ξT is ΛB,B′ ; the (B,B′)-

entry of ξS ′ is PB,B′ . Using 13.7(a) we deduce from 13.6(d) the equality of matrices

(a) ξM = (ξS)(ξT )(ξS ′).

As in the proof of 13.7(a) the last equality determines uniquely the matrices ξS, ξT ,ξS ′ if the matrix ξM is known; in fact, it provides an algorithm for computing theentries of these three matrices (and in particular for the entries PB′,B in terms of

the entries of ξM. Now under the bijection γ : B∼−→ ξB (see 11.12(a)) the matrix

ξM becomes a matrix indexed by B ×B whose (b, b′)-entry is v−κ(B)−κ(B′)(b, b′);these entries are explicitly computable from the combinatorial description of (:) onV. We see that:

(b) The quantities PB′,B are computable by an algorithm provided that the bijec-

tion γ : B∼−→ ξB is known.

This can be used in several examples to compute the PB′,B explicitly. Thealgorithm in (b) seems to depend on the choice of η such that η = δ; but in fact,by the results in [LY, 3.9], 10.22, 11.13, it does not depend on this choice.



In the case where m = 1, there is another algorithm to compute the quantitiesPB′,B ; see [L7, Theorem 24.8]. It again displays the matrix ξS as the first ofthe three factors in a matrix decomposition like (a), but with the matrix ξMbeing replaced by a matrix indexed by a pair of irreducible representation of aWeyl group and with entries determined by a prescription quite different fromthat used in this paper. In that case the bijection γ : B

∼−→ ξB is replaced bythe “generalized Springer correspondence”. It would be interesting to understandbetter the connection between these two approaches to the quantities PB′,B .

14. Odd vanishing

In this section we show that for any irreducible local system L on a G0-orbit in

gnilδ the cohomology sheaves of L� ∈ D(gδ) are zero in odd degrees. (See Theorem14.10.) In the case where m � 0, this follows from the analogous result in theZ-graded case in [L4]. In the case where m = 1 this follows from [L7, Theorem24.8(a)]. In the case where m > 1, δ = 1, L = Ql and G, g = ⊕igi are as in[LY, 0.3], this follows from [L8, Theorem 11.3] and from the known odd vanishingresult for affine Schubert varieties.

14.1. We preserve the setup of 10.1, 10.2. For � ∈ E′ let h(�) ∈ Z be as in 10.2.

For c ∈◦E we set h(c) = h(�) where � is any element of c∩E′; this is well defined,

by 10.7(a). For c ∈◦E we set Tc = v−h(c)Tc ∈ V′. Note that {Tc; c ∈

◦E} is a

Q(v)-basis of V′. Let f �→ f♥ be the field involution of Q(v) which carries v to−v; this extends to a field involution of Q((v)) (denoted again by f �→ f♥) givenby

∑i civ

i →∑

i ci(−v)i, where ci ∈ Q.Let b �→ b♥ be the Q-linear involution V′ → V′ such that (fTc)

♥ = f♥Tc for

any c ∈◦E and f ∈ Q(v).

Lemma 14.2 (14.2). For any b, b′ in V′ we have (b♥ : b′♥) = (b : b′)♥.

It is enough to show that for any c, c′ in◦E and any f, f ′ in Q(v) we have

(f♥Tc : f ′♥Tc′) = (fTc : f ′Tc′)♥.

We can assume that f = f ′ = 1. It is enough to show that

(a) (Tc : Tc′) ∈ Q(v2),

or thatv−h(�1)+h(�2)[�1|�2] ∈ Q(v2)

for any �1, �2 in E′ (notation of 10.12(a)) or that

(b) v−h(�1)+h(�2)∑w∈W

vτ(�2,w�1) ∈ Q(v2).

From 10.8(b) we see that for �,�′ in E′ we have

(b) τ (�,�′) = h(�) + h(�′) mod 2.

From 10.11(b) we see that for � ∈ E′, w ∈ W and N ∈ Z we have

dim εpw�N = dim ε

p�N , hence dim ε

uw�N = dim ε

u�N ,

so thath(w�) = h(�).



Using this and (b) we see that for �,�′ in E′ we have∑w∈W

vτ(�′,w�) ∈ vh(�

′)+h(�)N[v2, v−2].

Thus, (b) holds. The lemma is proved.

14.3. From 14.2 we deduce that ♥ : V′ → V′ carries Rl = Rr onto Rl = Rr;hence it induces a Q-linear involution V → V (denoted again by ♥). From 14.2 wededuce:

(a) For any b, b′ in V we have (b♥ : b′♥) = (b : b′)♥.

14.4. Let c ∈◦E, � ∈ E′′ ∩ c and let a� =

∑j v

−2sjvd(�) ∈ A be as in 11.1. Notethat

(a�)♥ =∑j

v−2sj (−v)d(�) = (−1)d(�)a�.

Hence we have

(a−1� Tc)

♥ = (a−1� vh(c)Tc)

♥ = (−1)d(�)+h(c)a−1� vh(c)Tc == (−1)d(�)+h(c)a−1

� Tc.

It follows that:(a) ♥ : V′ → V′ carries V′

A onto V′A and ♥ : V → V carries VA onto VA.

14.5. We show:a) For any b ∈ V′ we have b♥ = (b)♥. Hence for any b ∈ V we have b♥ = (b)♥.

We can assume that f = fTc where c ∈◦E and f ∈ Q(v). Note that f♥ = (f)♥.

Hence we can assume that f = 1. We have

Tc = v−h(c)Tc = vh(c)Tc = v2h(c)Tc,

hence

(Tc)♥ = v2h(c)Tc.

We have T♥c = Tc = v2h(c)Tc. This proves (a).

14.6. We show:(a) b �→ b♥ is a bijection B′ ∼−→ B′.Let b ∈ B′. From b ∈ VA we see using 14.4(a) that b♥ ∈ VA. From b = b we

see using 14.5(a) that b♥ = (b)♥ = b♥. From (b : b) ∈ Q(v) ∩ (1 + vZ[[v]]) we seeusing 14.3(a) that

(b♥ : b′♥) = (b : b′)♥ ∈ (Q(v)∩ (1 + vZ[[v]]))♥ = Q(v)∩ (1 + vZ[[v]]) ⊂ 1 + vZ[[v]].

Using this and the definitions, we see that b♥ ∈ B′. Thus the map b �→ b♥, B′ → B′

is well defined. Since this map has square 1, it is a bijection. This proves (a).

From (a) we deduce:

(b) If b ∈ B, then b♥ = sbb for a well-defined sb ∈ {1,−1} and a well-defined

b ∈ B.The following result makes (b) more precise.

Lemma 14.7 (14.7). Let O be a G0-orbit in gnilδ and let b ∈ BO. We have

b♥ = (−1)dimOb.



We argue by induction on dimO; we can assume that the result holds when Ois replaced by an orbit of dimension < dimO (if any). We have γ(b) = L�[dimO]

where (O,L) ∈ I(gδ). Let x ∈ O; we associate to x an ε-spiral pφ∗ and a splitting lφ∗

of it as in [LY, 2.9]. Recall that◦l

φ

η ⊂ O and that L1 := L|◦l

φ

η

is an irreducible Lφ0 -

equivariant local system on◦l

φ

η ; thus, L�1 ∈ D(lφη ) is defined. Let I = ε Indgδ

pφη(L�

1) ∈Q(gδ); we have clearly I ∈ ξQ(gδ). By [LY, 2.9(b),(c)], in ξK(gδ) we have

I = L� +∑

O′;dimO′<dimO

∑b′∈BO′

fb′γ(b′),

where fb′ ∈ A. Define I ′ ∈ VA by γ(I ′) = I. We have

(a) I ′ = v− dimOb+∑

O′;dimO′<dimO

∑b′∈BO′

fb′b′.

By [L4, 17.2, 17.3], in K(lφη ) we have

(b) hL�1 =

∑j∈J

hj indlφη

q(j)η(C(j)),

where h = h♥ ∈ A − {0}, J is a finite set and for j ∈ J , q(j) is a parabolic

subalgebra of lφη with Levi subalgebra m(j) such that q(j),m(j) are compatible

with the Z-grading of lφ, C(j) ∈ Q(m(j)η) is a cuspidal perverse sheaf and hj ∈ A.

Moreover, since L�1 belongs to the block of Q(lφη) given by ξ, we can assume that

m(j) = m and C(j) = C for all j. Thus (b) can be written in the form

(c) hL�1 = F0 + F1,

where

F0 =∑j∈J

h′j ind

lφη

q(j)η(C[− dimmη])),

F1 =∑j∈J

h′′j ind

lφη


h′j + h′′

j = hjvdimmη , h′

j ∈ Z[v2, v−2], h′′j ∈ vZ[v2, v−2].

Let K(lφη )ev be the Z[v2, v−2]-submodule of K(lφη )

ev with basis L′� for various

(O′,L′) in I (lφη ). By [L4, 21.1(c)], for any j ∈ J , we have

indlφη

q(j)η(C[− dimmη]) ∈ K(lφη )

ev.

Hence F0 ∈ K(lφη )ev, F1 ∈ vK(lφη )

ev. We have clearly hL�1 ∈ K(lφη )

ev. Since

vK(lφη )ev ∩ K(lφη )

ev = 0, we deduce from (c) that hL�1 = F0; that is,

hL�1 =

∑j∈J

h′j ind

lφη




where h′j ∈ Z[v2, v−2]. Applying ε Indgδ

pηand using the transitivity property 4.2

from [LY] (and also 10.4(a)) we obtain

hI =∑j∈J

h′jI�j

,

where �j ∈ E′ for j ∈ J . Since γ−1(∑

j∈J h′jI�j

) is fixed by ♥, it follows that hI ′

is fixed by ♥. Since h �= 0 and h♥ = h, we deduce that I ′♥ = I ′. Using (a) and theinduction hypothesis, we see that

(v− dimOb)♥ +∑

O′;dimO′<dimO

∑b′∈BO′

f♥b′ (−1)dimO′

b′

= v− dimOb+∑

O′;dimO′<dimO

∑b′∈BO′

fb′b′.

Using this and 14.6(b), we deduce that (−v)− dimOsbb − v− dimOb is a A-linearcombination of elements in ∪O′;dimO′<dimOBO′ .

Since b ∈ BO and b ∈ B this forces b = b and sb = (−1)dimO. This completesthe proof of the lemma.

14.8. We show:(a) Let O,O′ be G0-orbits in gnilδ and let B,B′ in B be such that the sup-

port of B (resp. B′) is the closure of O (resp. O′). We have (B : B′)♥ =

(−1)dimO+dimO′(B : B′).

If ψ(B) �= ψ(B′), then (B : B′) = 0 by [LY, 7.9(a)] and (a) holds. Assume nowthat ψ(B) = ψ(B′). We can assume that ψ(B) = ψ(B′) = ξ. It is enough to prove

that, if b ∈ BO, b′ ∈ BO′ , then (b : b′)♥ = (−1)dimO+dimO′

(b : b′). Using 14.2 and14.7, we have

(b : b′)♥ = (b♥ : b′♥) = ((−1)dimOb, (−1)dimO′b′)

and (a) is proved.

14.9. We show:(a) For any B,B′ in B we have PB,B′ ∈ N[v−2], PB,B′ ∈ N[v−2], ΛB,B′ ∈

Q((v2)) (notation of 13.6).The proof is similar to that of 13.7(a). With the notation in 14.8(a) we apply ♥

to

v− dimO−dimO′(B : B′) =

∑B1∈B,B2∈B

PB1,BΛB1,B2PB2,B′

(see 13.6(d)); using 14.8 we obtain

v− dimO−dimO′(B : B′) =

∑B1∈B,B2∈B

P♥B1,B

Λ♥B1,B2

P♥B2,B′ .

When B,B′ vary, this can be expressed as the decomposition of the matrix M =(vcB−cB′ v−κ(B)−κ(B′)(B : B′)) (indexed by B×B) as a product of three matrices

S♥T ♥S ′♥ where S♥ (resp. S ′♥) is the matrix indexed by B ×B whose (B,B′)-

entry is P♥B′,B (resp. P♥

B,B′) and T ♥ is the matrix indexed by B × B whose

(B,B′)-entry is Λ♥B,B′ . Recall from 13.6 that we have also M = ST S ′ (notation of

13.6). Thus we have

S♥T ♥S ′♥ = ST S ′.



Now by 13.6(c),(e), the matrix T (and hence the matrix T ♥) belongs to a subgroupof GLN (N = �(B)) of the form GLN1

× . . . GLNkwhere N1, . . . , Nk are the sizes of

the various subsets BO; moreover, by 13.6(a),(b), the matrix S (hence the matrixS♥) belongs to the unipotent radical of a parabolic subgroup of GLN with Levisubgroup equal to the subgroup of GLN considered above, while the matrix S ′

(hence the matrix S ′♥) belongs to the unipotent radical of the opposite parabolicsubgroup with the same Levi subgroup. This forces the equalities S♥ = S, T ♥ = T ,

S ′♥ = S ′. Now the equality S♥ = S implies P♥B′,B = PB′,B for all B′, B in B.

Similarly, from T ♥ = T we see that Λ♥B,B′ = ΛB,B′ for all B,B′ in B and from

S ′♥ = S ′ we see that P♥B,B′ = PB,B′ . This proves (a).

Theorem 14.10.(a) Let (O,L) ∈ I(gδ) and let A = L� ∈ Q(gδ). We have HaA = 0 for any odd

integer a.(b) Let � ∈ E′. We have Ha(I�) = 0 for any odd integer a.(c) Let (p∗, L, P0, l, l∗, u∗) be as in 4.1(a) of [LY] with ε = η, and let (O′,L′) ∈

I(lη) (see 1.2). We form A′ = L′� ∈ Q(lη). We have Ha(ε Indgδpη(A′)) = 0 for any

odd integer a.(d) In the setup of (c), ε Indgδ

pη(A′) is a direct sum of complexes of the form L�[s]

for various (O,L) ∈ I(gδ) and various even integers s.

(a) follows from 14.9(a).

We prove (b). Let c ∈◦E be such that � ∈ c. In V we have Tc = ⊕b∈Bfbv

−κ(b)b,where fb ∈ A. Applying ♥ and using T♥

c = Tc and (v−κ(b)b)♥ = v−κ(b)b for b ∈ B(see 14.7) we see that

⊕b∈Bf♥b v−κ(b)b = ⊕b∈Bfbv

−κ(b)b.

Hence f♥b = fb, that is, fb ∈ Z[v2, v−2]. Thus,

(e) I� is isomorphic to a direct sum of complexes of the form B[−κ(B)][2s] withB ∈ B and s ∈ Z.

HenceHaI� is isomorphic to a direct sum of sheaves of the formHa+2s(B[−κ(B)])with B ∈ B. Hence the desired result follows from (a).

We prove (c). By 1.5(a) of [LY] we can find q∗, (p∗, L, P0, l, l∗, u∗) as in 4.2 of [LY]

with ε = η and a cuspidal perverse sheaf C in Q(lη) such that A′[s′] is a direct sum-

mand of indlηqη(C[− dim lη]) for some s′ ∈ Z hence, by 4.2(a) of [LY], ε Indgδ

pη(A′)[s′]

is a direct summand of ε Indgδ

pη(C[− dim lη]); moreover, by [L4, 21.1(c)], we have

s′ = 2s′′ for some s′′∈Z. Hence Ha(ε Indgδpη(A′)) is a direct summand of

Ha−2s′′(ε Indgδ

pη(C[− dim lη])).

We can assume that pη is an ε-spiral with splitting m∗ (in 10.1) and that C = C(in 10.1). Then ε Indgδ

pη(C[− dim lη]) = I� for some � ∈ E′ and Ha(ε Indgδ

pη(A′)) is

a direct summand of Ha−2s′′I�. The desired result follows from (b).We prove (d). As in the proof of (c), ε Indgδ

pη(A′) is a direct summand of I�[−2s′′]

for some � ∈ E′ and s′′ ∈ Z. This, together with (e) gives the desired result. Thetheorem is proved.



References

[KL2] David Kazhdan and George Lusztig, Schubert varieties and Poincare duality, Geometry ofthe Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979),Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203.MR573434

[L4] George Lusztig, Study of perverse sheaves arising from graded Lie algebras, Adv. Math.112 (1995), no. 2, 147–217, DOI 10.1006/aima.1995.1031. MR1327095

[L6] George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes EtudesSci. Publ. Math. 67 (1988), 145–202. MR972345

[L7] George Lusztig, Character sheaves. V, Adv. in Math. 61 (1986), no. 2, 103–155, DOI10.1016/0001-8708(86)90071-X. MR849848

[L8] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math.Soc. 3 (1990), no. 2, 447–498, DOI 10.2307/1990961. MR1035415

[LY] G. Lusztig and Z. Yun, Z/m-graded Lie algebras and perverse sheaves, I, Represent. Theory21 (2017), no. 12, 277–321.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge,

Massachusetts 02139

E-mail address: [email protected]

Department of Mathematics, Yale University, New Haven, Connecticut 06511

E-mail address: [email protected]

http://www.ams.org/mathscinet-getitem?mr=573434





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